Hi Jorrie,

Great Post as usual. I have a couple of questions.

Is the rate of expansion of space between two stars, galaxies, etc. distance dependent? If so, how fast is that blackhole that created this light moving away from us now (since there is now 30 billion light years between us and it)?

Is there a redshift cutoff (z=20)? If so, why, if not, why not?

I thought about the question you raised, and I'm sure it involves the increasing rate of expansion and also involves when expansion exceeds c (and thus never reaches us), but I don't know how. Could you give an answer for that?

Thanks,

Roger

Hi Roger, you questions:

1. "Is the rate of expansion of space between two stars, galaxies, etc. distance dependent?"

For sure - that's the idea of the Hubble constant (71 km/s/Mpc). Only valid between galaxies on the large scale, not within local groups and clusters.

2. "If so, how fast is that blackhole that created this light moving away from us now (since there is now 30 billion light years between us and it)?"

-S has answered this below, using the cosmo-calculator.

3. "Is there a redshift cutoff (z=20)? If so, why, if not, why not?"

No, the optical cutoff lies at z~1080, where the cosmos became transparent. What we see in the CMB radiation is the 'surface of last scattering' at that redshift and time. No stars, no galaxies then - they are theorized to came into existence at around z=20. This is then a target distance for detecting and measuring redshift of galaxies, yet to be achieved.

-J

Thanks Jorrie. If it's not too much trouble, how do you calculate the optical cutoff?

Hi Roger, replying to your last two posts, asking:

1. "Why 5 billion years ago would the recession rate suddenly stop decreasing and start increasing?"

Theory has it that vacuum energy started to dominate about 5 Gy ago. Unlike other forms of energy density, it is not diluted by the expansion - vacuum energy density remains constant, while the density of the others, like mass energy, must necessarily reduce. It has this peculiar property that it 'blows up the balloon'. See my reply to Jon below. S also described it correctly is #13.

2. "If it's not too much trouble, how do you calculate the optical cutoff?"

It is the epoch of 'last scattering', when the temperature dropped enough (due to the expansion) to allow atoms to form and photons could move more or less freely for the fist time. That temperature is about 3000° K. Before that, the cosmos was opaque. The value of z ~ 1080 is calculated from the temperature, which is a function of the expansion history. It is also measured as the redshift of the CMB.

-J

Jorrie,

Thanks for your response. I was wondering.

1. In your first response you said that vacuum energy started to dominate about 5 GY ago because it is not diluted due to expansion (whereas gravity is). So here's the picture I have in my mind.

i) Beginning of the universe symmetries weren't broken and there was no higgs field, no mass, and no forces to resist vacuum energy expansion, thus very fast expansion ensued.

ii) As space expanded and things necessarily cooled (larger volume) , symmetry breaking occurred leading to massive particles (thanks higgs field) and forces. These new forces resisted the vacuum energy expansion (but did not stop it) leading to a slowing expansion.

iii) Five billion years ago expansion had reached a point where there was enough space now so that the vacuum energy could overcome the braking caused by gravity and the other forces, leading to an acceleration of expansion. (We are here now)

That's my understanding as of this minute. I guess my question is, is the time (5 Gy ago) that this change occur dependent upon the galaxy you are studying, or do you get the same time (5 Gy ago) for anything you look at (far enough away to be relevant). In other words, is this reference frame dependent?

2. I've heard this "there was less space and so the temperatures were much higher" statement before. I understand that classically that if you have a volume filled with say a gas, then constrict it, it gets higher in temperature, so in that sense it makes sense. Except, not really, because the gas molecules in that classical case collide with the molecules of the container in order to remain "contained". There is no "edge of space" that I'm aware or understand, so I though I understand the explanation, it seems kind of iffy to me. On the other hand, when you restrict a particles position, the uncertainty of it's momentum increases so I suppose that in a smaller universe, with lower uncertainty in (X,Y,Z) it could be argued that there is more uncertainty in (Px, Py, Pz). Is that the reason, or is there some other reason?

Roger

Hi Roger, your points 1. (i) to (iii) are essentially correct and in line with present mainstream thinking.

Your question: "... is the time (5 Gy ago) that this change occur dependent upon the galaxy you are studying, or do you get the same time (5 Gy ago) for anything you look at (far enough away to be relevant)."

Check figure 16.1 of Relativity 4 Engineers, shown here for reference. The change in acceleration happened gradually in curve (ii) from about t=5 to t=9 Gy, not suddenly. It is the whole curve (ii) that matters, not specific galaxies at specific distances. Curve (i) is for no vacuum energy and is a pure parabola.

It is true that the accelerating expansion was first noticed by looking at supernovae that happened some 10 Gy ago, meaning the light started out before the acceleration started to reverse polarity. But you obviously need data from a lot of distances (including the CMB) to be able to know the shape of the curve. Chapter 16 deals with the shape of the expansion in reasonable detail.

I'll have to think about your point 2. It's getting late in my valley - especially for a retiree.

-J

Hi Roger, on your point 2: "I've heard this "there was less space and so the temperatures were much higher" statement before.""

A universe that was more dense had to be hotter, not so? The "wall" that you refer to may have been there in terms of no more space for the particles to go to, so they just collided with each other and was very hot. But the cosmos expanded** and the temperature changed inversely with expansion factor a. The temperature at last scattering (CMB, with a~1088) was some 3000°K, and we measure that fairly directly. At this level, we do not need Heisenberg - although he may enter a higher density and temperature, closer to the BB.

** The high temperature was not the reason for cosmic expansion; blame quantum gravity(?) and vacuum energy for that.

-J

Thanks Jorrie. I'm still having trouble with the idea of the "space-nothing" interface, but I'm sure everyone does.

Hi Roger, some back-tracking^[1] to your first reply, where you wrote:

"I thought about the question you raised, and I'm sure it involves the increasing rate of expansion and also involves when expansion exceeds c (and thus never reaches us), but I don't know how."

Not quite correct.

For ease of reference, here is the question again: "BTW, can you figure why the blue D-then curve has its peculiar form - first increasing with z and then decreasing again? It actually approaches the horizontal axis asymptotically as redshift goes very large."

The hand-waving answer is perhaps the most enlightening. The light from high-z objects left them when the universe was very young and our present observable portion still very small. At that time all objects were very close to our cosmic location - in fact, at last scattering (CMB epoch) all particles were practically 'right on top of us'; well, not more than some 40 Mly away, which was the radius of our observable cosmos then.

The light from low-z objects left them not too long ago and what's more, they could not recede very far from their then-distance yet - given the short time and also the fact that 'nearby' items do not recede very fast today. 'Nearby' still means far enough today to have a proper Hubble flow that is not swamped by peculiar motions (generally z > 0.02, exceeding 300 million light years or so, e.g. the Coma cluster).

It should also be clear from the D-then equation on the web page Cosmic Distance Calculations.

-J

[1] I left it hanging for a while to see what else comes out. I haven't seen any others. Maybe I've missed some...

Hi Jorrie,

Interesting post as usual. I was a good idea to put the calculator here. I tried it out. The first thing that jumped out at me was the recession speeds. With 8.2 as you suggested, it showed 2.2c now and 3.4c then. This at first glance would indicate that the universe expansion is slowing down, and we both know modern theory says it is speeding up. If I remember correctly it slowed down for 8-10 billion years and then started speeding up, so I guess that explains it. Is my thinking anywhere close to being correct? Things are never obvious when it comes to the universe are they?

-S

Hi S, accurate thinking!

According to a rough calculation, that black hole's recession rate dropped from 3.4c to some 1.8c around 5 billion years ago and has since increased to 2.2c again. Its average recession speed is about 26.7/13 = 2.05c over the 13 billion years.

-J

You Wrote:"According to a rough calculation, that black hole's recession rate dropped from 3.4c to some 1.8c around 5 billion years ago and has since increased to 2.2c again."

Why 5 billion years ago would the recession rate suddenly stop decreasing and start increasing? Don't get me wrong, I believe you, I just don't understand why it would do that.

Hi Roger,

The change didn't happen suddenly. For the first few billion years the effect of gravity was dominant. As the universe expanded, that effect reduced, and the effect that is explained as the cosmological constant "push" (presently called dark energy) became dominant. Both forces as presumed to always have been there. There is some good general information about the universe here. Jorrie may have a graph that explains the expansion rate in his ebook, but I will let him answer that. Hope there are no hard feelings from previous threads. Certainly none on this end.

-S

p.s. see my current thread about my daughter.

Thanks Jorrie. I am very interested in this thread especially the direction you and Jon have taken it. I am also very confused - not much do to my current situation, I think. But I may be asking questions that are stupider than usual if that's possible. It's been some time since I submitted my Big Bang threads, and may have forgotten much.

• tell me what 1/a is and how it relates
• i'm baffled by the talk of comoving coordiantes
• peculiar motion I thought would be relative to "Hubble flow", but you seem to say it is relative to CMB?
• most of the tethering talk is confusing

With that said, I will go back to previous posts and ask questions and make comments as time permits.

-S

Hi S, some quick definitions/explanations:

1. The expansion factor a is simply the degree of expansion, linearly reckoned from a present a=1. The ratio 1/a is roughly equivalent to redshift z for large z (small a), since z = 1/a - 1 (by definition, z=0 for a=1).

2. Comoving coordinates is chosen so that distances equal the proper distances today (D-now). It is deemed to remain constant, in so far as that a galaxy with comoving distance 10 Gly was always 10 Gly (comoving) from us, even when it was created some 7 Gy ago, with a proper distance D-then of only 5.26 Gly. The 'comoving measuring rod' expands and/or shrinks with the universe. It's like a calibrated grid painted on an expanding balloon.

3. Relative to the Hubble flow and relative to the CMB are more or less equivalent. If one wants to split hairs, then it is only true if redshifts are corrected for our own peculiar movement, which can only be measured relative to the CMB.

4. The 'tethering' is just a way of saying that you take a nearby galaxy and hypothetically stop all proper movement relative to us. It is better to think that the galaxy is boosted in velocity towards us until there is zero redshift between it and us. This is not exactly the same thing as tethering, but it needs quite a bit of calculation to explain.

Hope it helps. Shout if I failed to make it clear enough...

-J

Thanks for the email nudge Jorrie. I like your calculator. Do you know any reason why it calculates slightly different results from the Wright and Morgan cosmo calculators, other than being accurate to more digits? I want to be as accurate as possible in my spreadsheet and I'm inclined to switch to your calculator since it apparently is.

The "D-then" line in your graph shows that the proper distance of photons from us that were emitted at high z progressively increased over long periods of time until the proper distance began decreasing at z ~ 1.6 as the photons cross our Hubble Sphere. By the same token, it shows that, for example, photons emitted at z=3 had a longer total proper propogation path to us than photons emitted at z=20. This effect is nicely illustrated in Figure 1.1 of Tamara Davis' 2003 doctoral thesis which is readily available on the web.

Although graphs of this type are well accepted in cosmology and are really helpful in understanding the complexity of cosmological redshift, I think one is justified in questioning whether this is the "real" story, since it requires that photons initially do propogate backwards relative to the eventual observer, in terms of proper distance (which is not a coordinate-specific measurement.) This is said to occur because the intervening space increases in volume spontaneously, with no identified cause, source, or mechanism for the ongoing volume increase.

You say:

"One must however not think of the recession rate as speed, because speed is an attribute of something that is or was accelerated. That black hole did not experience any appreciable acceleration; it is just that there is presently about 9 times more space between the black hole and us than when it was born. This is the very essence of cosmic expansion."

We discussed this point at great length many months ago, and I'm still holding out. I can't accept the characterization that although the distance between a galaxy and us is increasing, no "movement" is in fact occurring. Neither GR nor the FLRW metric seem to me to require that characterization. The many published analyses of the "tethered galaxy" thought experiment make it perfectly clear that neither galaxies nor particles are being "pushed" apart by "expanding space" (with the possible exception of dark energy). The simplest conclusion then is that galaxies actually are moving apart from each other, with a residuum of plain vanilla momentum left over from inflation (or the big bang), decelerated constantly over time by mutual gravitation, and reaccelerated more recently by the "push" of the cosmological constant.

Likewise, the wave peaks of propogating photons are not pushed apart by the intrusion of newly created space between the peaks. The residual flow of the cosmic fluid isn't exerting some metaphysical mystery force on the wave peaks propogating through it. So although cosmological redshift happens to occur exactly in proportion to the expansion of the universe since emission (making the expanding space explanation seem so obvious), the expansion itself cannot be the cause of the redshift. Rather, the redshift must result from the complex (and partially offsetting) interactions of relativistic Doppler effect resulting from the emitter's recession velocity "then", and gravitational time dilation arising from the cosmic density dynamics at emission time and during propogation. At least I haven't seen any analysis that rules out such a proposition.

Jon

Hi Jon, good to hear from you again.

You asked about the differences between my 'Hellfire' variant and the others. AFAIK, it lies in the radiation energy component being left out by Wright and Morgan. I think they intended theirs for galaxy redshifts in the range z = 0 to 10, where radiation energy density is quite negligible.

You wrote: "By the same token, it shows that, for example, photons emitted at z=3 had a longer total proper propagation path to us than photons emitted at z=20."

I suppose it depends on how you define "total proper propagation path". If we integrate proper motion of a photon, we get the lookback distance, which shows the opposite effect. A photon received with z=3 has moved ~11.5 Gly, while the one with z=20 has moved ~13.5 Gly through its local (proper?) space.

You further said: "The simplest conclusion then is that galaxies actually are moving apart from each other, with a residuum of plain vanilla momentum left over from inflation (or the big bang), decelerated constantly over time by mutual gravitation, and reaccelerated more recently by the "push" of the cosmological constant."

The problem is, how do you reconcile the superluminal recession rates and "plain vanilla momentum" with GR? The 'surface of last scattering' (CMB) must have been receding at ~65c and galaxies with z=20 at ~6.2c from us. Moving through space? Nah!

-J

Aha! So what can explain such superluminal recession rates, AND account for the cosmological redshift? This is the heart of the dilemma that must be solved. I suggest we start by assuming there IS a valid answer other than "mystery expansion". I have an idea but I am can't nail down the math yet.

Regarding the total propogation path of a photon, I agree there is more than one way to look at it. But you will agree that in proper global coordinates, the NET distance "D-Then" from an emitter at z=3 is longer than the NET distance "D-Then" from an emitter at z=20, as your chart illustrates. That's what I was referring to, without regard to the photon first moving backward and then retracing its path.

Jon

Actually I'm not at all sure there is a valid solution that avoids the need for expanding space. But I hope there is, because the "tethered galaxy" results seem to flat-out contradict the idea that the volume of space continues to grow spontaneously as a result of some "initial condition" 14+ Gy years ago and is progressively prying galaxies, particles and wave peaks away from each other. Peacock published a "diatribe" about the contradiction but for some reason it doesn't get anywhere near the publicity that topics such as "dark energy" and "dark matter" do as one of the most astounding mysteries/contradictions of cosmology.

Jon

I see that the radius of the observable universe is 46.6 Gly. I was Googling and found that the diameter of the complete universe is calculated to be 156 Gly as shown here. That's only about 50% bigger. I seem to recall you saying it was a million times bigger. Am I still thinking accurately?

Hi StandardsGuy,

Wikipedia in the "Misconceptions" section of its entry "Observable Universe" points out that the 156 Gly figure is wrong:

"156 billion light-years. This figure was obtained by doubling 78 billion light-years on the assumption that it is a radius. Since 78 billion light-years is already a diameter, the doubled figure is incorrect. This figure was very widely reported."

It references that Space.com article as one of the sources of this error.

Clearly 78 Gly is just the diameter which is twice the 49 Gly radius Cornish arrived at in his CMB study.

Jon

Hi S, as Jon has said in #9, that article had it wrong. It was just a faulty report of the proper radius of the observable universe, which in 2004 was thought to be ~39 Gly instead of the ~47 Gly today.

You wrote: "I seem to recall you saying it was a million times bigger."

I don't recall the "million times bigger" for the total universe, but more like that there is a theoretical/observational minimum size which is about ten times the observable universe of today (my blog entry "How BIG is the Cosmos?") . You may be remembering that the universe expanded about 1000 times since "last scattering", which produced the CMB.

Present data indicates that our universe could be anything upwards from 400 Gly radius to infinite in size, because the "flatness factor" (curvature parameter) includes slightly positive curvature and precisely flat.

-J

Hi Jorrie,

There's a slight discrepency in the way your cosmic calculator generates "Recession speed then". For example at z=1023, when I convert your "Hubble parameter then" to Ly/y/Ly in my spreadsheet (using an exact conversion factor) and multiply it by your "Proper distance then", I calculate 62.9930c instead of the 63.0228c "Recession speed then" your calculator generates. (I didn't change any of the starting parameters.)

Do you know what causes the discrepency?

Jon

Jorrie:

Also, at any high z the "Proper distance of the source now" times Ho of 71 converted to Ly/y/Ly does not quite equal the "Recession speed of the source now" generated by the calculator.

Jon

Hi Jon, I've got some catching up to do, so I'll reply to your last posts together.

1. "... because the "tethered galaxy" results seem to flat-out contradict the idea that the volume of space continues to grow spontaneously as a result of some "initial condition" 14+ Gy years ago and is progressively prying galaxies, particles and wave peaks away from each other."

I fail to see how this contradicts the standard model. If a galaxy is "tethered" it is taken out of the Hubble flow and is actually forcefully accelerated through space, say towards us. If the expansion rate drops, it will fly towards us and eventually go away in the opposite direction. If the expansion is accelerating, it will stay at a constant proper distance from us (because the tether will remain taut). Stock-standard LCDM cosmology - people resisting the "expanding space" concept have been pretty quiet lately...

I try to keep my sanity by thinking that Lambda (vacuum energy) is 'blowing up the balloon', operating in an extra dimension. This is more or less what Georges Lemaître 'preached' in 1933, when he found the first important inhomogeneous solution to Einstein's field equations. His 'cosmic egg' was just the inflationary epoch, also caused by Lambda. The fact that we do not understand what vacuum energy is should not deter us more than our inability to understand why the 'quantum recipe' works...

2. "There's a slight discrepency in the way your cosmic calculator generates "Recession speed then". For example at z=1023, when I convert your "Hubble parameter then" to Ly/y/Ly in my spreadsheet (using an exact conversion factor) and multiply it by your "Proper distance then", I calculate 62.9930c instead of the 63.0228c "Recession speed then" your calculator generates."

That's in fact very, very close - much closer than the errors in the parameters! E.g. H₀ is not known to better than ~3%. However, I think the closeness is coincidental, because the conversion from redshift to recession speed cannot be done by simple conversion and multiplication, except at very low z. It is labeled the "naive Hubble calculation", I think.

3. "Also, at any high z the "Proper distance of the source now" times Ho of 71 converted to Ly/y/Ly does not quite equal the "Recession speed of the source now" generated by the calculator."

Ditto. It is a non-linear solution that can only be obtained out of the full numerical integration.

-J

Hi Jorrie, you said:

"However, I think the closeness is coincidental, because the conversion from redshift to recession speed cannot be done by simple conversion and multiplication, except at very low z. It is labeled the "naive Hubble calculation", I think."

Sorry, I don't think I communicated clearly. The "naive" Hubble calculation refers to the method of trying to calculate recession velocity directly from redshift, c*z/Ho. That method becomes wildly inaccurate unless z << 1.

That's not what I was talking about. I started by using your calculator to generate the "proper distance then" and the "hubble parameter then." (Or, "proper distance now" and Ho of 71.) I'm sure your calculator uses the comoving distance integration formula featured on your web page, so it should be accurate. I'm sure you also calculate the Hubble rate using the omega density formula, which is exact but does not involve integration.

When your "H" is multiplied by your "D", the answer should be the exact recession velocity. That's the "true" Hubble's law which does not involve any relativistic adjustments. What other method would one use to accurately calculate recession velocity?

I suspect that your recession calculation differs slightly from mine because you use the Ho/980 Hubble conversion factor for Ly/y/Ly. I think that's a convenient approximation but it's not as accurate as the exact conversion factor I use.

Jon

Hi Jon, OK, I now see what you mean with the slight differences.

I actually rounded to H_bar=Ho/978, c=300 000 km/s, with 3.26 ly/parsec in the calculator. Barring round-off errors, the calculator converts v_rec = H x D correctly, as far as I checked.

E.g. at z=1000, it gives D_then = 45.5314 Mly, H_t = 1 334 834.48 km/s/Mpc and v_rec = 45.5314 x 1 334 834.48 / (3.26 x 300 000) = 62.1441c.

How does your calc differ?

-J

Using your H_t and your D_then I calculate v_rec at 62.1147. (You have 62.1441).

I use the following to calculate the H_ly/y/ly conversion factor:

c = 299,800 km/s

ly = 9,460,730,472,580.8 km

mpc = 3,261,564 ly

H_ly/y/ly conversion factor = H_t * 1.02201208689542E-12

The conversion factor of course is just (km/ly) / (sec/yr) / (ly/mpc).

And, your 20,000 integration steps sounds accurate enough for D_then!

Jon

Hi Jon, it seems to me that our differences are simply round-off errors in our constants and also in our calculators.

Point in case is that if I put your exact constants for c and mps/ly into my Exel spreadsheet, I actually get v_rec = 62.1557.

-J

Strange. Did you try using my exact conversion factor for H_ly/y/ly? I'm also using Excel.

Jon

Hi Jon, you wrote: "Strange. Did you try using my exact conversion factor for H_ly/y/ly? I'm also using Excel."

No, I'm not using that conversion factor at all. My calculation is a straight: v_rec = 45.5314 x 1 334 834.48 / (3.261564 x 299800) = 62.1557c.

It may well be rounding errors in that tiny factor of yours that cause the differences. Does it help to have so many counting digits after the decimal point and then shift the decimal point 12 places to the left? The digital system can't handle that many digits, I think.

In any case, the small differences are probably irrelevant. I've just entered the latest WMAP data as defaults into the cosmo-calculator and the results are somewhat different anyway! Check the edited note 4 of the opening post, where you can run the old calculator with the new one in two windows for comparison.

-J

Hi Jorrie,

OK I figured out the rounding occurred just because my approach used more math steps than yours. When I streamlined it down, I get very close to your figure. I think some rounding occurs with every step, so the fewer the better!

Also, it would be great if your calculator could include a calculation of total density (rho).

Jon

Hi Jon, further to my reply #22 to your question on the small discrepancies:

Perhaps the major cause of small differences is the number of steps in the numerical integration loop. The present cosmo-calculator uses 20,000 equal steps for a=0 to 1. If I up the number of steps by a factor 10, the now-distance and recession rate go up by some 0.07% (32 Mly and 0.0023c). The 20,000 steps seem to be a good trade-off for speed/performance, especially if we remember that parameter uncertainties may cause some ±5% deviations anyway!

-J

Hi Jorrie, you said:

"If a galaxy is "tethered" it is taken out of the Hubble flow and is actually forcefully accelerated through space, say towards us. If the expansion rate drops, it will fly towards us and eventually go away in the opposite direction. If the expansion is accelerating, it will stay at a constant proper distance from us (because the tether will remain taut).Stock-standard LCDM cosmology - people resisting the "expanding space" concept have been pretty quiet lately..."

People have not been quiet about this subject lately, I think that at the moment the panel is a bit stumped about the nature of the very real contradiction of expanding space, and most of the readily apparent points have already been articulated.

Prof Peacock apparently is still befuddled by the contradiction, as he makes clear in his 9/08 update to his paper http://arxiv.org/abs/0809.4573v1 "A diatribe on expanding space." Referring to the Ω=1 example he says "The 'expanding space' idea would suggest that the test particle should indeed start to recede from us, and it appears one can prove this formally.... [applying the 1/a decay rate of peculiar velocities] ... Now for the paradox.... The acceleration is negative, so the particle moves inwards, in complete contradiction to our 'expanding space' conclusion that the particle would tend with time to pick up the Hubble expansion.... In no sense, therefore, can 'expanding space' be said to have operated.... This analysis demonstrates that there is no local effect on particle dynamics from the global expansion of the universe: the tendency to separate is a kinematic initial condition, and once this is removed, all memory of the expansion is lost.... The earlier examples have proved that 'expanding space' is in general a dangerously flawed way of thinking about an expanding universe."

In the above passage Peacock raises an interesting point about memory of the expansion being lost once a particle is diverted from the Hubble flow. Consider an extention of his thought experiment, in which the tethered galaxy is brought to a halt (relative to the observer's galaxy) and then is immediately untethered and re-accelerated by rockets back up to some recession velocity (radially away from the observer.) The rockets are turned off at some location where the recession velocity (relative to the observer) of the galaxy happens to be equal to the appropriate local Hubble rate, thereby puting the galaxy "at rest" with the local Hubble flow in its new neighborhood. The galaxy will from then on behave exactly like a "natural" galaxy in the Hubble flow and it will observe galaxies all around it receding from it in accordance with the normal Hubble law (Of course, the galaxy's own final location in comoving coordinates relative to the observer will be different from its location prior to the initial tethering). The galaxy's recession rate relative to the observer will not decay over time at the 1/a rate applicable to peculiar motions, because we've already caused the galaxy to have a net zero velocity in comoving coordinates (it's at rest in its local comoving hubble flow). So in Peacock's terms, the galaxy may have "forgotten" the Hubble flow, but we've just "re-taught" it. This shows that peculiar motion is in fact kinematically equivalent to expansion motion at least locally.

It should be self-evident that the 1/a decay rate for peculiar motion does not apply when a particle with non-zero peculiar velocity (relative to the local Hubble flow) is decelerated such that its peculiar velocity relative to its new local Hubble flow becomes zero. One must think of "peculiar velocity" not in terms of HOW a particle gained that velocity, i.e. whether it originated from the gravitational effect of non-homogeneous mass distributions or other forces (such as man-made forces), but strictly in terms of whether a particle is currently exactly at rest with its local comoving Hubble flow.

The "peculiar motion" of a particle can be thought of simply as a temporary perturbation such that (regardless of the cause) the particle finds itself in a perturbed comoving coordinate LOCATION relative to its state of motion. There is always some location in the universe (let's call it "destination X") such that if a particle with peculiar velocity were instantly transported there (without changing its momentum relative to its location before being transported), its peculiar velocity would become zero because its velocity is now in synch with (comoving with) the local Hubble flow there. And elegantly, the 1/a decay rate means that the momentum of every particle with a peculiar velocity will seek to carry the particle across comoving coordinates directly toward its own "destination X" until it eventually comes asymptotically to rest (i.e. comoving with the local Hubble flow) at that exact spot.

You said:

"I try to keep my sanity by thinking that Lambda (vacuum energy) is 'blowing up the balloon', operating in an extra dimension."

Yes I agree that that analogy works well for Lambda, but that's not the contradiction I'm talking about. I am referring to the residual expansion left over from inflation (or the BB), without regard to Lambda. As you know, the original expansion operates exactly like momentum in the sense that, in the absence of Lambda acceleration or gravitational deceleration, galaxies would continue to separate at an absolutely constant rate forever. And the effect of mutual gravitation is thought to act on receding galaxies in exactly the same way it would act on any massive test particles placed in uniform motion away from each other. The balloon analogy doesn't work here unless you can find a balloon that continues expanding forever after we stop blowing air into it (assuming no external force or energy is applied.)

Jon

Hi Jon,

"There is always some location in the universe (let's call it "destination X") such that if a particle with peculiar velocity were instantly transported there (without changing its momentum relative to its location before being transported), its peculiar velocity would become zero because its velocity is now in synch with (comoving with) the local Hubble flow there."

So, do you picture the matter in the universe as a 'bomb' that exploded into infinite space? The idea of expanding space has been around for a long time. I'm not sure what the evidence is for that opinion other than the acceleration expansion (which is relatively new). There is the CMB and the relative isotopes issues, but would they also support this model? (Jorrie jump in here?) If you believe this, then you would not believe inflation I presume.

-S

Hi S.

My only problem with Jon's statement that you referenced is that although correct, the process is not reversible. What would happen if a far-away particle, with recession rate of 2c, "were instantly transported [here] (without changing its momentum relative to its location before being transported)"? Will it be super-luminous here?

It may point to a problem with the way in which Prof. Peacock analyzes the cosmos at large.

-J

Hi StandardsGuy,

I don't have any firm opinion about how the universe started out, because there don't seem to be enough facts yet to make a judgment, just lots of variations on theories.

I do tend to think that ever since a time shortly after the Big Bang (or after inflation ended), the expansion of space occurred by means of "real" movement of galaxies rather than by means of expanding space (or expanding hypersurface as Jorrie terms it). So I think it expands kinetically like an explosion, whether or not it actually started out as one. Of course when you lay Lambda (dark energy or the cosmological constant) on top of the recent expansion, it adds an additional dynamic.

I have no firm view on whether the universe is infinite or is finite and has an "outer edge". Philosophically I prefer finite, because infinity is hard to get one's head around. Either answer is consistent with observations to date, as long as we are located far enough away from any outer edge that it doesn't show up in our observation of the CMB. If we were located at least 1100 times further from the edge than the present radius of our observable universe, the CMB should show no traces of edge effects. If an edge effect were observed, at first blush it might look to us like a cold spot in the CMB.

Jon

Hi Jon.

On Prof. Peacock's issues with expanding space - I don't quite understand why he has a problem. My favorite 'balloon analogy' does exactly what he describes and my space expands!

Consider an Einstein-de Sitter balloon (flat, matter only) with us and one medium distance galaxy, pictured as an insect on the balloon, receding at low redshift. If the insect starts to crawl towards us at a constant local rate that just cancels out its (then) redshift, it will eventually pick up a blueshift and will hence reach us, as can be modeled very simply. Remember the Einstein-de Sitter expansion rate eventually approaches zero.

The insect never picks up the 'Hubble flow' of this balloon - Peacock's analysis of that must be very funny (or flawed). If the insect suddenly stops crawling towards us (coming to rest in the CMB frame again, far enough away not to be gravitationally bound to us), it will immediately pick up the local Hubble flow again - just like Peacock's galaxy that's been rocketed back to its 'proper' redshift. So, where's the problem?

In a realistic LCDM universe balloon analogy, the increasing expansion rate will make even the crawling insect to pick up the Hubble flow over time (minus a tiny peculiar velocity towards us). It still does not mean that the Moon ever picks up the Hubble flow! The model says nothing about 'local space'.

On your problem with my balloon analogy: my balloon surface has mass/radiation/vacuum-energy and it has momentum in the super-dimension perpendicular to the surface. When Lambda stopped blowing air into the balloon (after inflation), the surface coasted outwards kinematically, with the surface energy causing a deceleration due to mutual gravitational pull. It hence appears as if all galaxies drift apart on the large scale. Today Lambda is again blowing into the balloon, we think...

Being too philosophical about the reality of an expanding hyper-surface is futile, IMO. All we can do is create models and adjust them to fit observations.

-J

Hi Jorrie,

You said:

"Consider an Einstein-de Sitter balloon (flat, matter only) with us and one medium distance galaxy, pictured as an insect on the balloon, receding at low redshift. If the insect starts to crawl towards us at a constant local rate that just cancels out its (then) redshift, it will eventually pick up a blueshift and will hence reach us, as can be modeled very simply. Remember the Einstein-de Sitter expansion rate eventually approaches zero.

The insect never picks up the 'Hubble flow' of this balloon - Peacock's analysis of that must be very funny (or flawed)."

No, if you think about it a little more you'll realize that the ant's motion on the expanding balloon is not at all the same as the purely kinematic motion that would occur if the balloon weren't expanding.

The ant's total travel distance (and elapsed time) will be significantly longer on an expanding balloon than on a non-expanding balloon. And if the balloon's initial expansion rate exceeds the ant's maximum local speed, the ant will find itself progressively farther away from us for a period of time, until the balloon's expansion rate eventually slows down enough for the ant to be able to overtake it and begin approaching us. Obviously that wouldn't happen on a non-expanding balloon. Travel on the surface of an expanding balloon is fundamentally different from that on a non-expanding balloon, regardless of whether the expansion is decelerating or accelerating. This is exactly why peculiar motion through expanding space is fundamentally different from peculiar motion through non-expanding space.

Consider the scenario where the ant on an expanding balloon is marching directly away from you. The ant's constant "peculiar motion" away from you (in comoving balloon coordinates) will progressively decay at the rate of 1/a, and eventually the ant will be asymptotically at rest with you in comoving coordinates. Contrast that with a non-expanding balloon, where the ant's peculiar motion away from you will never decay in comoving coordinates, because the comoving Hubble rate is zero. Confusion arises if you try to mix proper distances with comoving distances, which are apples and oranges.

You said:

"When Lambda stopped blowing air into the balloon (after inflation), the surface coasted outwards kinematically, with the surface energy causing a deceleration due to mutual gravitational pull."

Agreed! Regardless of what initial conditions caused the original expansion, the ongoing separation of massive particles ever since then has been caused 100% by truly kinematic coasting (conserved momentum of the massive particles themselves), and 0% by ongoing "push" from expanding space (again, disregarding Lambda). Surely you're not suggesting that empty vacuum is like a river that retains "momentum" over time and carries captive massive particles along with it.

Jon

I said:

"Consider the scenario where the ant on an expanding balloon is marching directly away from you. The ant's constant "peculiar motion" away from you (in comoving balloon coordinates) will progressively decay at the rate of 1/a, and eventually the ant will be asymptotically at rest with you in comoving coordinates."

Also consider the same scenario where an ant at some distance from you is marching away from you, but then is "tethered" to your position on the surface. On an expanding (but decelerating) balloon, upon release the ant will accelerate (skid) toward you on the slippery surface of the balloon (assume no friction), pass right through your location, and continue skidding away in the opposite direction.

Conversely, on a non-expanding balloon, upon untethering the ant will never skid toward you no matter how slippery the surface is.

This thought experiment isn't entirely clear because realistically an ant moves by continual propulsion (application of force) against the drag of friction, as distinguished from the cosmological scenario where there is no friction and the "ant" in freefall is a passive body which applies no ongoing force to effectuate its movement.

Jon

I said:

"Also consider the same scenario where an ant at some distance from you is marching away from you, but then is "tethered" to your position on the surface."

Oops, upon further reflection, I completely confused myself in describing this scenario. The situation is different and more complicated than I described.

The non-expanding balloon needs to have a force of gravity, which has no obvious parallel on a balloon. (It's not equivalent to a shrinking balloon). Maybe a magnet is the easiest analogy, although the magnet needs to decline in power over time (to represent the decreasing gravitational density of the cosmic fluid).

So on the non-expanding balloon there is a magnet at your location, pulling on the ferric ant. When the ant is untethered, it is the magnet that pulls the ant toward you and through your location. The ant's momentum carries it indefinitely far beyond your location at a decaying velocity (in both proper and comoving coordinates). Your magnet is constantly losing force and so has less effect on the ant after it passed you than it had while the ant was accelerating toward you.

Conversely on the expanding but decelerating balloon, it's unclear what the ant's motion would be due to the uncertainties of "balloon physics." The ant has a comoving velocity toward you, but the expanding surface is trying to carry him away from you. If the surface is very "sticky", (i.e. the expansion of space applies irresistable force on massive objects) the ant will immediately rejoin the Hubble flow away from you. If the surface is very slick (i.e. the expansion of space applies no force on massive objects), I think the ant will behave somewhat the same as on a non-expanding balloon. But I think the latter is inconsistent with the underlying physical model of how expanding space supposedly affects massive bodies, which is the irresistable force model.

I think there's a limit to how much learning one can squeeze out of the balloon analogy, but it models certain aspects of expanding space really well.

I think it's pertinent to ponder whether the expansion of space could possibly have an immediate effect on particle motions, causing essentially instantaneous acceleration of extremely massive objects away from the origin. F=Ma, so that requires almost infinite force. If expanding space is so "unyielding" and "non-compressible" in that sense, then how are we able to so easily force peculiar motion through space at all? Space should be essentially impenetrable.

Jon

Hi Jon, you wrote: "No, if you think about it a little more you'll realize that the ant's motion on the expanding balloon is not at all the same as the purely kinematic motion that would occur if the balloon weren't expanding."

I disagree - relative to the balloon surface, the expansion or not does not matter. Locally light moves at c and the ant moves at whatever constant speed it can muster. The fact that it has to travel farther on an expanding balloon to reach us does not matter here.

On the kinematic momentum, you wrote: "Regardless of what initial conditions caused the original expansion, the ongoing separation of massive particles ever since then has been caused 100% by truly kinematic coasting (conserved momentum of the massive particles themselves), and 0% by ongoing "push" from expanding space (again, disregarding Lambda). Surely you're not suggesting that empty vacuum is like a river that retains "momentum" over time and carries captive massive particles along with it."

In my mind, the hyper-surface has the motion and momentum (in the hyper-radial direction), while the comoving galaxies have zero momentum relative to the hyper-surface. Whether this means the creation of more empty space to accommodate the larger spatial surface area, or whether it means a stretched surface, is immaterial to me. Except perhaps that more empty space is easier to reconcile with the total vacuum energy that grows with the expansion (constant vacuum energy density). I know this may run into certain problems for a perfectly flat and infinite universe, but then I just make my hyper-radius approach infinity as well.

BTW, I may perhaps reverse your closing question and ask: Surely you're not suggesting that massive particles flow through the empty vacuum like free particles through static water?

-J

Hi Jorrie,

Well you're right, I should back off on using the term "expanding space" as if it were a perjorative term. I think your term "expanding hypersurface" works well and doesn't carry as much baggage.

I'd like to focus your attention on one argument I made in opposition to the expanding hypersurface model. Although it may sound pedantic or even silly, I think the logical linchpin here is something I'll nickname the "hypersurface coefficient of friction," just to make it easier to discuss.

Let's start with this question: If you let a ball drop onto an (infinitely long) moving conveyor belt, upon impact will the ball roll backwards (away from the direction of the belt's movement), in "comoving belt" coordinates? That is, if you paint stripes across the belt at 1cm intervals, will the ball roll backwards from the impact point across some number of stripes, until it comes to (comoving) rest on the belt? The answer is:

(a) No, if the belt's surface is divided by crosswise vertical barriers of appropriate size that prevent the ball from rolling at all. Think of this as the belt having a 100% coefficient of friction.

(b) Yes in perpetuity, if the belt's coefficient of friction is zero. At the limit, zero friction is equivalent to the ball never making contact with the belt at all (e.g., the ball is suspended by air jets at a mm above the belt's surface.

(c) Yes at a decelerating rate for a limited time, if the belt's surface has no cross barriers and has a "normal" coefficient of friction (somewhere between 0% and 100%.) The ball will start moving backwards, slow down and eventually stop.

Next question: If we place some massless test particles in a large, non-expanding spherical hollow shell pattern in homogeneous (Lambda=0) space, will the expansion hypersurface (if there is one) cause the particle shell to start expanding outward (in proper distance) at the Hubble rate Ho, or at less than Ho? The latter is equivalent to the ball rolling backward on the conveyor belt. The answer is:

(a') The particle shell will expand outward instantly at exactly Ho, if the hypersurface's coefficient of friction is 100%.

(b') The particle shell won't expand at all, and in fact will accelerate inward, if the coefficient of friction is 0. The shell particles will accelerate toward the center (due to the background cosmic gravitation), and if the particles just miss colliding with each other at the center, they will all pass through the center and thereafter keep expanding outward, in an inside-out shell configuration, for at least as long as the universe continues to expand.

(c') If the coefficient of friction is less than 100% but greater than 0, the shell will initially accelerate inward like case (b'). Then, depending on how the coefficient of friction compares to the changing deceleration factor q of the background cosmic gravitation, the shell particles may eventually stop, reverse course, pass back through the center, and then expand outward (in a right-side out configuration) up to a maximum velocity that could asymptotically approach Ho.

Considering this logic, the tethered galaxy analysis assures us that the coefficient of friction of the expanding hypersphere must be zero. The other two scenarios just don't fit the expansion metric. Which means that the hypersphere is unable to gain any "traction" on any particles to cause them to move in any manner. If it can't get traction on particles with peculiar motion, then it can't get traction on particles that happen to be comoving with the Hubble flow either. So if there is such an expanding hypersphere, the best one can say is that it is comoving along with the dust particles and galaxies, but it is not what makes them move. If the expanding hypersphere isn't causitive, arguably the concept should just be discarded.

Of course it's still possible that the expanding hypersphere actually did engage with mass-energy in the BB era, and it was the "initial condition" that put all of the mass-energy into recession motion. But while it may have had traction under the special conditions of inflation, it lost that traction the instant inflation ended.

And one might try to employ the expanding hypersphere as a sort of explanation for Lambda. I.e., the kinematic momentum of the dust particles alone suffices to keep the original expansion going (at a decelerating rate), but the hypersphere is expanding in parallel with the dust particles and is slowly accelerating, applying a weak sort of pressure on the dust particles that causes the dust particles to stop decelerating and start accelerating. But a theory like that seems inconsistent to me: why should the expanding hypersphere have a zero coefficient of friction as long as the expansion is dominated by mass-energy, and then transition to a 100% coefficent of friction when Lambda dominates? It doesn't seem to explain or simplify anything.

Anyway, disregarding Lambda, it should be clear that the tethered galaxy analysis flatly contradicts the theory that an expanding hypersurface is the mechanism which causes the continuation of the original tendency of galaxies to separate.

Jon

Hi again, Jon.

I'm not going to work through your whole post, because my model has no "hypersurface coefficient of friction" in any normal spacetime direction. One may perhaps call the total gravitational influence that slows down expansion (in the absence of Lambda) a "coefficient of friction", but this works in the hyper-radial direction. We must be careful not mixing the hyperspherical (balloon) model with your kinematic movement (through space?) model, because they do not mix very well.

Your: "So if there is such an expanding hypersphere, the best one can say is that it is comoving along with the dust particles and galaxies, but it is not what makes them move. If the expanding hypersphere isn't causative, arguably the concept should just be discarded."

Yea, but the hypershere makes the particles "move" in the hyper-radial direction. It is the hypothetical cause of the effect that we observe: more and more proper distance appears between particles, without them moving one bit in normal space, i.e. relative to the CMB. How's that for defending against your "discard notion"?

BTW, it seems to me that the latest WMAP data suggests that the 'flat' model may be ruled out now and that the universe is just closed, with k_eq = 0.00968±0.00046. If confirmed, this will be a blessing, because the infinite cosmos is then ruled out. Also, I like it, because the hyperspherical model will work just fine!

-J

I'm not mixing the "expanding hypersphere" model with the kinematic model. I'm trying to demonstrate to you step by step why your "expanding hypersphere" model predicts particle behaviors that violate the FLRW metric, as demonstrated by the tethered galaxy analysis.

If you won't bother to work through my explanation (which I put a lot of effort into), then we'll just consider this part of the dialogue to have ended with an agreement to disagree.

Thanks for the discussion. As you know, I'm always up for a good arguing session!

Jon

You wrote: "I'm trying to demonstrate to you step by step why your "expanding hypersphere" model predicts particle behaviors that violate the FLRW metric, as demonstrated by the tethered galaxy analysis."

Jon, likewise, I'm honestly trying to be patient in order to show you that the 4-d hyperspherical model fully conforms to the FLRW metric.

The only part that I refused to comment on point by point is the "hypersurface coefficient of friction," and "conveyor belt" concept of post #44. I'll prefer to leave that as is.

I still enjoy the robust arguments, Jon, so please don't go away.

-J

Hi Jorrie,

Well despite my suggestion that we agree to disagree, I'll comment on specific points you made.

You said that one may suggest that gravitation has a "coefficient of friction". Obviously if one did that, gravitation would have a "coefficient of friction" of 100%, because 100% of the gravitational accelerative force gets transferred to the particles it affects. That happens in the hyper-radial direction as well as in all other directions. In this sense, the predictions of gravitational effects are well behaved in both proper and comoving coordinate systems.

You say your "expanding hypersphere" has no coefficient of friction (I think you say that just to belittle my terminology), but then you contradict that statement by saying the hypersphere makes particles "move" (without changing their comoving coordinates) in the hyper-radial direction. How can the hypersphere cause any particle to become relocated "in normal space" as you call it (i.e. in proper distance) if the hypersphere can't "get a grip" on the particle, in other words if it has a zero coefficient of friction? In that case it couldn't have any effect on the particle's proper location. Conversely, if the hypersphere has a 100% coefficient of friction (which is what I think you really mean), then it does grip particles and cause their positions to change in both proper and comoving coordinates. In which case the hypersphere should also grip particles that have peculiar motions, and cause a change in their coordinate motion as well. But the FLRW metric analysis (tethered galaxy analysis) says that doesn't happen. Your expanding hypersphere does not cause a particle with peculiar motion to alter its trajectory in either the hyper-radial direction or in any other direction, in either proper or comoving coordinates. So unlike gravity, the predictions of your expanding hypersphere are not well behaved in either proper or comoving coordinates.

You imply that there is something quasi-magical or metaphysical about comoving coordinate systems that makes them fundamentally different from proper coordinate systems. There isn't. It's straightforward to translate between particle motions across coordinates on the one hand, and the expansion of the comoving coordinates themselves on the other hand. So when you say that the hypersphere model doesn't mix well with the kinetic (proper distance) model, I don't know where you're coming from. Of course particle behaviors (as functions of coordinate distance and time) can be translated with exact precision between proper and comoving coordinates. And in either coordinate system, one must be able to accurately portray the behavior of an individual particle which is either at rest or in motion relative to dust particles that are comoving with the local Hubble flow. One just needs to be clear which description of a particle's behavior corresponds to which coordinate system.

I think we've exhausted the ant on a balloon analogy and should discuss this subject in terms of specific particle behaviors in proper and comoving coordinate systems.

Jon

calculation of the average
density of space or dark matter
per cubic centimeter....gram
confusion
almost to abstract,,,,to impossible

near to perfection....................
a vacuum
in between,,,,,
a delay of space

on a small "scale""
some contrast
A difference between units
divide up the range of temperatures into equal.................C^0

but now
the real thing...
the big "scale""

the whole design
inside the print
change????

The GALAXY concentrate...
with the the leading note...............
fraction between the distance...............
density,I believe philippe martin<>

will get denser

but harmonic
uniform
and so relax................

UNIVERSE
and HER..
motion...........

ignoring inertia.............

uniform
without
paradox......

perfect,and so full
of
contrast....

the atomic nucleus
belong to the beginnings
the
EXPANSION<>

the elementary particle
smooth
and strategically......................

distinguish
HER self
PLUS,MINUS

erotic form..........
of your matter

huge ENERGY
density,

exponential expansion,
without.
paradox.

cosmology repulsion
balancing HER self...............................
GRAVITY...................................................philippe martin

I wrote: " ... because my model has no "hypersurface coefficient of friction" in any normal spacetime direction. One may perhaps call the total gravitational influence that slows down expansion (in the absence of Lambda) a "coefficient of friction", but this works in the hyper-radial direction."

I now regret writing this, because it is clearly already misinterpreted, i.e. your: "... gravitation would have a "coefficient of friction" of 100%, because 100% of the gravitational accelerative force gets transferred to the particles it affects. That happens in the hyper-radial direction as well as in all other directions."

In the homogeneous hyper-spherical model, the comoving dust particles all move only in the 4th spatial dimension and has no movement in the normal spatial directions. They stay at the same comoving coordinates for ever, unless something push them out of position. In the homogeneous case there are no gravitational gradients that can do any pulling, hence no gravitational collapses.

You: "Conversely, if the hypersphere has a 100% coefficient of friction (which is what I think you really mean), then it does grip particles and cause their positions to change in both proper and comoving coordinates."

Wrong. Zero changes in comoving coordinates, just in proper distance coordinates, due to the expanding hyper-surface.

You: "In which case the hypersphere should also grip particles that have peculiar motions, and cause a change in their coordinate motion as well. But the FLRW metric analysis (tethered galaxy analysis) says that doesn't happen."

Where does it say that? Only in the very special case of an empty universe (0,0) with a peculiar motion exactly canceling recession velocity relative to the origin. (BTW, the hyper-spherical model predicts exactly the same). In all realistic case, e.g. (1,0) or (0.3,0.7), both models predict that objects with peculiar velocity will change both proper and comoving coordinates. And they do not need gravitational pull to do so!

You: "You imply that there is something quasi-magical or metaphysical about comoving coordinate systems that makes them fundamentally different from proper coordinate systems. There isn't."

Where did you get this "quasi-magical or metaphysical " impression from? I simply use the coordinate system that is most appropriate for the problem at hand. Of course they are mutually convertible.

You: "So when you say that the hypersphere model doesn't mix well with the kinetic (proper distance) model, I don't know where you're coming from."

It is your version of the "kinetic (proper distance) model" that I have problems with. E.g. your view of why the untethered galaxy 'falls' towards the origin is apparently the gravitational influence of the mass density around the origin. Davis says on pdf page 59: "To enable us to isolate the effect of the expansion of the Universe we assume that the galaxies have negligible mass." The author considers only the various expansion dynamics as the cause. It is such differences in your view that makes it incompatible with the hyper-spherical model (which is compatible with Davis, BTW).

You: "I think we've exhausted the ant on a balloon analogy and should discuss this subject in terms of specific particle behaviors in proper and comoving coordinate systems."

Isn't that what we are doing, apart from not discarding the hyper-spherical model, because I won't be happy with that. I will also stick to isotropic, homogeneous solutions only and avoid the voids like the swain flu virus...

-J

Hi Jorrie,

Sorry to see you have lost a GA. You must have made somebody mad. I think I got 2 of mine by making somebody mad! I'm confused again/still. You told Roger that I described it right in 13 where I said "Both forces are presumed to have always been there." Now:

"When Lambda stopped blowing air into the balloon (after inflation), the surface coasted outwards kinematically, with the surface energy causing a deceleration due to mutual gravitational pull. It hence appears as if all galaxies drift apart on the large scale. Today Lambda is again blowing into the balloon, we think..."

I think inflation was something separate from Lambda. In any case this makes it look like an in-continuity. Furthermore, I will restate what Jon said in different words (but possibly a different meaning than he intended): Are you suggesting that empty space (which has no mass) has momentum?

Your reference to "at last scattering" is the point in time when the universe changed from opaque to transparent?

In the reference to LCDM(0.3,0.7), the 0.7 is Lambda? A (0,0) universe is one without a cosmological constant?

Consider a balloon partially blown up. Two buttons are glued to it. A string is tied between the buttons (the tether). Now the balloon is blown up some more, The tether is taut. When the tether is cut, the buttons fly apart, not come together, not so?

-S

Hi S. Frankly, I don't care too much about GAs.

You asked: "Are you suggesting that empty space (which has no mass) has momentum?"

I suppose if we could find completely empty space, with no energy at all, no virtual particles, no radiation, etc, it would have no momentum. Practically, space always has all these things and expanding space must hence have momentum.

Where Jon and I differ, I think, is that I view expanding space (with its mass, radiation and other energies) to have momentum only in the hyper-spherical (4th) spatial dimension. Further, I view only things that move through space to have momentum in the 3-D sense. My impression is that Jon reckons that things going with the Hubble flow has momentum in 3-D space, without actually moving through space.

"Your reference to "at last scattering" is the point in time when the universe changed from opaque to transparent?" Yes.

"In the reference to LCDM(0.3,0.7), the 0.7 is Lambda? A (0,0) universe is one without a cosmological constant?"

Yes - actually, the 0.7 is Ω_Λ, the ratio of vacuum energy density to the critical energy density, the latter meaning a 'flat' cosmos.

On your 'tethered buttons': the 'cosmic buttons' are not glued to the balloon, but sitting frictionless on the surface. Once given a 2-D spatial (relative to the balloon surface) momentum towards each other by the taut tether and the expanding balloon, their behavior when the tether is cut will depend on the expansion law of the balloon. This touches the main argument between Jon and myself.

If the expansion rate of the balloon remains constant, the buttons will remain at a constant proper distance from each other (exactly the original length of the tether). This represents the (0,0) case.

If the balloon is being blown up faster and faster (accelerating expansion), the buttons will drift apart, as measured in 'tether lengths' (which is equivalent to proper distance in cosmology). This is equivalent to the (0.3,0.7) LCDM case.

It then follows that if the balloon is being blown up slower and slower (decelerating expansion), the buttons will move closer to each other, as measured in 'tether lengths'. This represents the (1,0) CDM case.

I hope this helps you to follow the 'great debate', of which the outcome is far from certain!

-J

Hi Jorrie,

Thanks for your clear explanations.

On the 'empty' space, yes there are virtual particles (I don't like the name much), and so 'potential' mass, but I see no reason for that to cause momentum. Is this your view only, or that of Prof. Peacock or some other? I've never heard of this before, but not something I want to argue about.

I'm unable to side with you or Jon on your argument without reading the papers you are discussing. I have not the mental fortitude to read any right now. In fact I had a physical headache after reading one of Jon's posts a couple of days ago. At some point I may want to read them.

I see no point in discussing hypothetical tethering and un-tethering. In fact I can see no reason for there to be any tethering at all. Who started this idea? Give Roger my regards for his post 36. He seems to have done some reading.

-S

Hi S, on "empty space": it is accepted that vacuum energy makes up ~72% of the expansion energy of the present cosmos. This implies that "empty space" must have momentum, a very slippery concept! I get my head around it by means of the hyperspherical model. Although the 4th spatial dimension may be hypothetical, it gives very convenient mathematics for expressing the momentum of empty space - it is "energy on the move" into the extra dimension.

On the tethered galaxy: I think prof. Peacock first dreamed it up as a thought experiment in order to demonstrate problems with the concept of expanding space, if incorrectly applied. As your "buttons on the balloon" thought experiment showed, it is a good pedagogical tool. Also, as the arguments between Jon and myself illustrate, it is not a trivial problem to solve. I have a suspicion that Jon and I might both be right, i.e., it may just be two different mathematical models giving identical final results, but the jury is still out...

-J

"it may just be two different mathematical models giving identical final results"

A good possibility. I had the same thought as I have read various things about the 'Big bang' in recent years. They all describe it in some mathematical model that they have learned. There may be twists and turns, but the same basic result remains. This also occurred to me in our previous argument about velocity vs. acceleration in GR. I'm not enough of a mathematician to show my case.

"on "empty space": it is accepted that vacuum energy makes up ~72% of the expansion energy of the present cosmos. This implies that "empty space" must have momentum,"

If I understand you correctly you are saying that the other 28% of the expansion energy is momentum. I hadn't thought of that.

"Although the 4th spatial dimension may be hypothetical, it gives very convenient mathematics for expressing the momentum of empty space - it is "energy on the move" into the extra dimension"

I know that English is not your first language, so I think you meant in instead of into, but you didn't mean only the 4th dimension is expanding? When you blow into a balloon, the 2 dimensions (X and Y) get expanded otherwise the dots painted on the balloon would not move apart.

Hi S, your: "If I understand you correctly you are saying that the other 28% of the expansion energy is momentum."

No, if we throw away the 28% that is matter, the vacuum energy quanta will still have momentum in the hyperspace dimension, because this is where they are moving in. Momentum is defined as energy moving through space and since the vacuum energy quanta do not move through 3D space, they do not have 3D spatial momentum. Neither have galaxies going with the Hubble flow.

This is one place where the balloon analogy can be misleading - unless one realizes that the 3rd balloon dimension must be viewed as a hyper-dimension, invisible to the beings on the balloon surface. And that momentum means movement relative to the molecules of the balloon surface in the immediate vicinity of the object under consideration.

-J

PS: yes, the language of the Brits is not my mother tongue. Americans fortunately understand our "Afrikaner-English" better than what the Brits do! I think the reason is that we read/watch a lot more books/TV/movies from America than from Britain.

"the 'cosmic buttons' are not glued to the balloon, but sitting frictionless on the surface. Once given a 2-D spatial (relative to the balloon surface) momentum towards each other by the taut tether and the expanding balloon, their behavior when the tether is cut will depend on the expansion law of the balloon."

This and the following paragraphs are rather hard to swallow. I can't see that they would be frictionless. To me, without a tether they will "have" to follow the expanding space. With a tether and with no friction, the tether would not be taut, as there would be no force on it. The buttons would move apart regardless of the acceleration of the expansion (expansion is still going on). So I find myself agreeing with Jon more than you based on logic alone.

-S

S, you wrote: "I can't see that they would be frictionless. To me, without a tether they will "have" to follow the expanding space. With a tether and with no friction, the tether would not be taut, as there would be no force on it."

Remember that "untethered" in this thought experiment means that the buttons were tethered and then the tether is cut. If identical, the buttons will both have momentum towards each other relative to the 2D balloon surface, caused by the tether, precisely canceling the "moving apart causing by the expansion of the balloon. As I described in the paragraphs following (that you referred to), what happens next depends on whether the expansion rate of the balloon is constant, decreasing or increasing.

Jon and myself agree on this, just on the mechanism causing the effect. IMO, the literature that we referenced also agree with how I have explained it.

-J

OK, I think I am getting the picture. Each galaxy can be considered "in tow" by the other, giving them momentum. When the tether is cut, they still have that momentum toward each other. At first I thought you were discussing galaxies not expanding with the Hubble flow that was discussed before. So I had pictured each star tethered to each other inside a galaxy. I had forgotten to ask clarification. Now I understand it is only a thought experiment.

-S

Hi again Jon, some thoughts on your: "Prof Peacock apparently is still befuddled by the contradiction, as he makes clear in his 9/08 update to his paper http://arxiv.org/abs/0809.4573v1 "A diatribe on expanding space.""

I have problems with his (or someone else's?) 1/a decay rate for peculiar motion, which causes the 'paradox' that he describes. If peculiar motion is defined as motion relative to the CMB frame, then it remains constant for a fee particle (not gravitationally influenced), as far as I can establish. One can also postulate a series of comoving (fundamental) observers measuring the local speed of the inertial particle as it passes them in turn. This must remain constant, not so?

I think it is only when peculiar motion is defined in terms of comoving distance that it shows this 1/a decay issue over time. A hand-waving equivalent argument is that if space expands 'underneath' a particle, its local velocity must drop over time. I don't buy this as real, since I think local momentum must be conserved.

-J

Hi Jorrie,

First, I agree with you that the 1/a decay rate of peculiar motion is easy to misconstrue. Clearly it is a feature unique to comoving coordinates. In proper coordinates the motion decays (if at all) only at the deceleration factor q as determined by gravity and Lambda. E.g. in the standard model with Lambda, proper velocity accelerates even while comoving peculiar velocity decreases.

Comoving coordinates are useful for some purposes, but I think they've caused a lot of confusion in cosmology because they introduce a level of abstraction which tends to obscure simple behaviors and makes them seem metaphysical.

Second, I disagree with your statement that a free particle's motion is not influenced by the cosmic background gravity. This was a point that I had great trouble getting my head around at first. But eventually Wallace convinced me that one must apply the Newtonian shell theorem (Gauss' Law).

In a homogeneous gravitational dust background (with no Lambda, for simplicity), define an observer at the coordinate origin, and a massless test particle coasting away at radius r. Treat the sum of all dust mass within the sphere defined by r as a single point mass centered at the origin, and then calculate the gravitational force the point mass exerts on the test particle. That tells you how the test particle's motion will decelerate relative to the coordinate origin.

The weird thing is that in a homogeneous cosmos, one can define a coordinate origin anywhere one wants for this purpose and it still works. The shell theorem applies to every such arbitrary sphere because this deceleration is exactly the cosmic deceleration factor q everywhere, and every location in the universe experiences the same collapse action (deceleration) toward every other location. Every dust particle displays the same kind of decelerative motion as the test particle does, relative to any arbitrary origin. Every dust particle has residual motion left over from [inflation], and every particle experiences the same deceleration in the direction of every other particle.

If you think about it, this is exactly the reason why the untethered galaxy not only moves toward the origin (when Lambda = 0), but actually accelerates toward it. (And as I mentioned, once the galaxy passes through the origin, it accelerates back toward the origin, but that backward acceleration is weaker than the original acceleration because the comic density has decreased since the test began.)

This same decelerative cosmic collapse action also contributes the gravitational blueshift which the equivalence principle tells us should be one of the components making up the aggregated cosmological redshift (along with relativistic Doppler shift, and...)

Jon

Hi Jon. It seems Wallace has swayed your opinion, but I still have difficulty with that.

"In a homogeneous gravitational dust background (with no Lambda, for simplicity), define an observer at the coordinate origin, and a massless test particle coasting away at radius r."

I assume this dust background is infinite and that the test particle is not really massless, like a photon? I presume it just has negligible mass. What you describe is a local inhomogeneity, rendering the LCDM model useless there. Restore homogeneity, even roughly, and the test particle will roughly retain any peculiar velocity it has relative to the homogeneous background.

Your: "If you think about it, this is exactly the reason why the untethered galaxy not only moves toward the origin (when Lambda = 0), but actually accelerates toward it."

Do you mean the galaxy is tethered for while and then untethered? Then I do not need Wallace's inhomogeneity introduction to show that the galaxy will fall towards the origin - the tether gave it a pull in that direction! In the RW coordinate system (with perfectly homogeneous mass distribution) centered on us, it will appear as if it accelerates towards the origin, even with no net gravitational influence on it. Due to simple Hubble flow, it will start of with near zero blueshift, but the blueshift will grow as it passes comoving points en-route. My favorite balloon analogy explains this perfectly.

Your: "This same decelerative cosmic collapse action also contributes the gravitational blueshift ..."

Again, this is only valid once inhomogeneities are introduced. We, sitting inside a galaxy, observe all incoming light with an inherent gravitational blueshift, subtracting from the cosmological redshift. IMO, it has however no bearing on the large scale cosmological effects.

-J

Hi Jorrie, you said

"What you describe is a local inhomogeneity, rendering the LCDM model useless there. Restore homogeneity, even roughly, and the test particle will roughly retain any peculiar velocity it has relative to the homogeneous background."

This argument is a red herring. I'm assuming a scenario where any inhomogeneity is vanishingly small. For example, the test particle is a single Hydrogen atom and the radius is 1 megaparsec. At that scale, the mass of the background dust in the sphere defined by the radius absolutely dominates over the mass of the particle.

Don't mix up proper coordinates and comoving coordinates. In comoving coordinates, a free particle's peculiar velocity will decay at 1/a. In proper coordinates, the particle's velocity will never change at all except as a result of the accelerative effect of the cosmic background dust field's gravity. (Again assuming Lambda=0.)

You misunderstand the tethered galaxy results. At the instant the the tethered galaxy is untethered, it has zero proper velocity toward the origin. In the absence of any accelerative force, it would just sit there at a constant proper distance forever. That's nicely illustrated in Fig 3.2 (p.62) of Tamara Davis' thesis paper for a (0,0) empty universe. Figure 3.3 shows the same set of scenarios in comoving coordinates, which of course show the (0,0) particle moving inward (obviously because zero proper velocity = Ho inward peculiar velocity in comoving coordinates.) In the proper distance chart Fig 3.2, the obviously huge difference between the empty (0,0) model results and the Ω=1 (1,0) model results clearly confirms that the gravitation of the cosmic background is the cause, and the only cause, of the untethered galaxy accelerating inward.

You are also wrong that there isn't a large scale cosmological blueshift component in a homogeneous Ω=1 universe. This is explained in section 3 of Peacock's paper.

I can cite other peer reviewed papers that make these same points.

Jon

Hi Jon, you wrote: "This argument is a red herring. I'm assuming a scenario where any inhomogeneity is vanishingly small."

Recall that you wrote: "Treat the sum of all dust mass within the sphere defined by r as a single point mass centered at the origin, and then calculate the gravitational force the point mass exerts on the test particle. That tells you how the test particle's motion will decelerate relative to the coordinate origin."

Treating the "sum of all dust mass within the sphere defined by r as a single point mass centered at the origin" creates a huge inhomogeneity! That's what I was referring to, not the mass of the test particle (which obviously can't be massless, as you wrote, but can be of negligible mass). Anyway, I dispute the notion that the test particle will accelerate due to gravitational influences of a homogeneous cosmic scale dust, other than following the Hubble flow.

You: "In proper coordinates, the particle's velocity will never change at all except as a result of the accelerative effect of the cosmic background dust field's gravity."

It depends on what you mean by 'proper coordinates' and 'the cosmic background dust field's gravity.' I don't follow what you mean, for the same reason as above.

You: "At the instant the the tethered galaxy is untethered, it has zero proper velocity toward the origin. In the absence of any accelerative force, it would just sit there at a constant proper distance forever. That's nicely illustrated in Fig 3.2 (p.62) of Tamara Davis' thesis paper for a (0,0) empty universe."

True for an empty universe, but not so for a matter-dominated universe, where the untethered galaxy starts to move towards the origin (as Peacock has proved) and for a Lambda-dominated universe, where it starts to drift away from the origin. I've read Tamara Davis (2004) quite some time ago and I recall nothing that conflicts with what I wrote, at least for a realistic universe. A quick scan of the figures also confirms this.

Further you said: "In the proper distance chart Fig 3.2, the obviously huge difference between the empty (0,0) model results and the Ω=1 (1,0) model results clearly confirms that the gravitation of the cosmic background is the cause, and the only cause, of the untethered galaxy accelerating inward."

No! It is simply a function of decelerated, constant or accelerated expansion. In a homogeneous, matter-dominated universe, there are no gravitational effects other than the large-scale deceleration of the expansion. Davis wrote: "We have shown that an object with a peculiar velocity does rejoin the Hubble flow in eternally expanding universes but does not feel any force causing it to rejoin the Hubble flow."

You: "You are also wrong that there isn't a large scale cosmological blueshift component in a homogeneous Ω=1 universe. This is explained in section 3 of Peacock's paper."

I think you are interpreting Peacock incorrectly. Where he wrote: "If we think of the observer as lying at the centre of a sphere of radius r, with the emitting galaxy on the edge, then the sense of the gravitational shift will be a blueshift: ...", it is because he opted to relate large cosmological redshifts to standard Doppler shift, in which case he is forced to bring in gravitational red- or blueshift as well. I think this is an unncessary complication, simply because Peacock seems to abhor 'expanding space'. Embracing expanding space explains cosmological redshift without reverting to such antics.

In reality, there are no gravitational blueshifts of photons traveling in a homogeneous universe, because each observer is in the center of her own observable sphere and at exactly the same gravitational potential. Gravitational red- or blueshifts can only happen between points of different gravitational potential, where clocks run at different rates.

-J

Hi Jorrie,

A "quick scan" of the Davis paper obviously isn't sufficient for you to understand your misinterpretation. I urge you to study Fig 3.2 and 3.3 more carefully. You are so dug in that apparently the only way to approach this problem is in baby steps. So I won't respond to all of your comments.

You asked what I mean by "proper coordinates." Of course I mean coordinates that show proper distance on one axis and proper time on another, just like the coordinates in Davis' Fig 3.2.

You asked what I mean by "the cosmic background dust field's gravity". I defined a dust filled Ω=1 FLRW universe, and I'm talking about that comoving dust, and the fact that, uncontroversially, it does gravitate. Usually when I use this term I'm referring to the sphere of dust between the origin and distance r.

Now can we agree on one thing: The tether brings the tethered galaxy into a trajectory such that, at the instant of untethering, it begins the exercise at ZERO PROPER VELOCITY toward the origin? (Yes it has inward COMOVING velocity, but zero PROPER velocity.)

Can we agree that ZERO PROPER VELOCITY means it ain't moving, compared to the origin, in the normal sense of those words, at t=0?

But surprise, surprise, when we look at the (1,0) trajectory in Tamara's Fig 3.2, immediately upon being untethered, the galaxy ACCELERATES inward toward the origin, in proper coordinates. That's why the line slopes to the left. It wasn't already moving, it was standing still. (Continuing to stand still would display in the chart as a vertical line.) And now it is moving. That change is the definition of ACCELERATION, the second derivative of proper distance. (Notice the tiny little curvature in the (1,0) line just at the "now" line. The large scale of the chart tends to de-emphasize the curvature.)

SOMETHING caused the galaxy to accelerate inward. What was that something? Did the "expanding hypersurface" cause it to accelerate inward? No, that doesn't make any sense, if anything an expanding hypersurface would impart outward acceleration to the galaxy.

Let's see, by process of elimination the only remaining accelerative force candidate is --- (drum roll please) ---- GRAVITY !

You keep talking about the untethered galaxy "moving" inward as if it had a pre-existing proper velocity. It doesn't! Its initial proper velocity (the first derivative of proper distance) is ZERO. A particle can't change from being stationary to moving inward just because of some mystical counter-expansionary effect of "expansion." Again, the only way to change from zero proper velocity to non-zero proper velocity is to ACCELERATE.

This is not a difficult concept.

Jon

Jon, sorry, I'm not going to respond to your post #53, because I think my previous few posts covered it already. I have quoted two small portions of Davis 2004 in my last few posts - it is perhaps time you reread those sections in context and contemplate what they really tell you.

-J

Jorrie, I can't continue this discussion until you respond to my post #53. It highlights a key point you are missing.

I can (and eventually will) respond your out of context quotes from Tamara Davis, but right now that line of discussion will only distract you from post #53 which needs to be the starting point.

In post #53 I try to focus your attention on the obvious fact that the tether's only purpose is to bring the galaxy to zero proper velocity relative to the origin. So upon release of the tether, in the first instant the galaxy is not moving relative to the origin. Tamara Davis' Fig 3.2 shows that at that point, the tethered galaxy begins accelerating toward the origin. That's why the (1,0) line immediately veers off to the left (unfortunately you practically need a magnifying glass to make out the curvature.) That change in the (1,0) line from vertical to a leftward slope signifies inward acceleration. Please consider and respond to this point in terms of proper velocity, not comoving velocity.

A chart which shows much more explicitly the curvature in the proper trajectory caused by inward acceleration is fig 2 on p.5 of this paper by Barnes, Francis, James & Lewis "Joining the Hubble Flow: Implications for Expanding Space." There's lots of other good discussion in that paper, but please let's just focus on the inward acceleration point for now.

You will note that the authors of that paper refer frequently to the "Newtonian solution." Newtonian meaning gravity.

A good place to read more about the inward gravitational acceleration caused by an arbitrary sphere of comoving dust is this paper, "Cosmological Radar Ranging in an Expanding Universe" by Lewis, Francis, Barnes, Kwan & James. Particularly the discussion at the end of section 4 on p.5. Again, please just focus on the topic about inward acceleration in proper velocity. We can talk about the other stuff later.

Jon

Hi Jon, you wrote:

"Jorrie, I can't continue this discussion until you respond to my post #53. It highlights a key point you are missing."

I thought I've given you the broad arguments many times now; so it looks as if I will also, reluctantly, have to revert to the "baby steps", as you called it. If you happen to prove me wrong, I'll be delighted, because I would have learned something.

Before we go there, just a remark on your: "I can (and eventually will) respond your out of context quotes from Tamara Davis, ..." Recall that I asked you to reread the referred-to sections in order to establish the correct context, but let's debate that later.

1: "So upon release of the tether, in the first instant the galaxy is not moving relative to the origin." Agreed for proper distance coordinates.

2. "That change in the (1,0) line from vertical to a leftward slope signifies inward acceleration." Agreed that there is a change in proper velocity. The issue is, why?

3. "Please consider and respond to this point in terms of proper velocity, not comoving velocity." In a sense, I have to consider both to make the "why" more understandable - it is obviously not very obvious, else we would not still have been arguing.

I will first give my view of why the untethered galaxy moves (except in the (0,0) case) in the absence of any local gravitational gradient and then why there can be no gravitational gradient in a perfectly homogeneous cosmos, as required by the FLRW model. We can debate what the literature says later.

3.1 The tether has given the galaxy a negative velocity relative to a local comoving observer located at its starting location - call this the local velocity, local relative to every comoving observer that the galaxy will pass. It is not quite a comoving velocity, because it does not diminish over time.

3.2 The cosmic expansion rate (da/dt) continues to drop for the (1,0) universe, even if only asymptotically as t -> infinity.

3.3 This means that the expansion rate at the starting location at and every location between the starting location and us also drops.

3.4 Due to this drop, the constant negative local velocity starts to outstrip the expansion rate, causing the proper distance to the galaxy to decrease.

3.5 A decreased proper distance causes the expansion rate at the new location of the galaxy to drop even further. It is a double-whammy and the galaxy quickly (on cosmological timescales) picks up a sizable velocity in proper distance coordinates. This explains the "sudden" bends in Tamara Davis' Fig 3.2 to some degree.

3.6 All this happens without considering local gravity. We only consider the mutual gravity of the matter in the universe at large that slows down the expansion rate. Tamara Davis' Eq. 3.6 confirms that.

3.7 In the (0,0) case, the expansion rate does not change; hence there is no proper movement for the galaxy in question. In the LDCM (0.3,0.7) universe, given that the expansion rate is already increasing, the reverse of the above process essentially happens and the galaxy quickly picks up a positive proper velocity, receding from us.

4. Now onto the local gravity issue:

4.1 The FLRW model requires either an infinite or a finite, closed, unbounded universe, which is homogeneous and isotropic in both cases.

4.2 In such a universe the gravitational potential is the same everywhere and there are no gravitational gradients.

4.3 To be FLRW compatible, your cosmic dust must comply with the same condition.

4.4 I think the Shell Theorem only works for a finite, bounded dust cloud (or body) that has a center of mass. It is used in the modeling of the collapse of dust and dark matter, seeded by the inhomogeneities that we observe in the CMB. It is not applicable to the homogeneous universe at large.

4.5 If we replace the particles of the infinite or unbounded cosmic dust with evenly spread galaxies, then stars (inside and outside the galaxy) will experience gravitational gradients.

4.6 However, the evenly spread galaxies themselves will feel exactly the same 'gravitational pull' from every direction and it cancels out - the same as saying there are no gravitational gradients on a cosmic scale.

4.7 Hence, the untethered galaxy does not fall towards any origin under the influence of gravity – its peculiar motion and the expansion dynamics alone cause it. This is why Tamara Davis (2004) says on pdf page 59: "The motion of this test galaxy reveals the effect the expansion of the Universe has on local dynamics. To enable us to isolate the effect of the expansion of the Universe we assume that the galaxies have negligible mass." I can quote other papers, but Davis is sufficient.

I hope this drives us closer to an agreement on the primary issue!

-J

PS: I've updated the cosmo-calculator with a "density-then" box, as you suggested.

Hi Jorrie,

Good, we agree on your #1 and #2: At the instant of untethering, the galaxy has zero proper velocity toward the origin, and then its proper velocity inward begins to accelerate. What caused it to accelerate?

3. You are cheating! This analysis is to be done in terms of proper velocities only!

3.1 I think you're just saying that the galaxy has a peculiar velocity inward. I agree that its velocity is peculiar relative to the untethered galaxy's local Hubble flow at its own location, but it is not peculiar relative to the origin's local Hubble flow. Relative to the origin, its peculiar velocity is the same as its proper velocity: zero. If the peculiar velocity relative to the origin is zero, then by definition the galaxy cannot cross comoving coordinates in order to reach the origin, which is at a different comoving coordinate.

3.4 The drop in the background Hubble velocity does not cause the galaxy's proper velocity to change relative to the origin. The proper velocity started at zero, and it won't change from zero unless a "force" acts on it. You are not distinguishing between comoving and proper motion here, so you're coming up with a mishmash (albeit a clever mishmash).

If the proper velocity of a particle that starts with peculiar motion (relative to the background Hubble flow) actually increased as the Hubble rate drops (or even if the proper velocity remained constant), then peculiar velocities would not decay at exactly 1/a in comoving coordinates, the math would calculate a different result. So we know that any such notion is conclusively wrong. In the tethered galaxy problem, the galaxy's comoving velocity does not decay at 1/a while the galaxy approaches the origin, because the galaxy's proper velocity is accelerating. Only after the galaxy passes the origin will the rate of decay in the peculiar velocity begin to approach 1/a.

3.5 In the absence of gravitational acceleration, the proper distance from the origin would never change, so this point is moot.

3.6 Tamara Davis' Eq 3.6 just takes advantage of the fact that the decrease in a particle's peculiar velocity is equal to the decrease in the Hubble velocity (because the same gravitational deceleration parameter q is working against both), so one can work backward from the changing Hubble velocity (which is a proper velocity) and the scale factor (which is proportional to a proper distance) to calculate the changes in the particle's proper distance. The equation doesn't tell us what causes the proper velocity to change.

Tamara goes on to say (p.47): "Thus all terms in [dot R] cancel and we conclude that the expansion, [dot R] > 0 does not cause acceleration, [double dot D] > 0. Thus, the expansion does not cause the untethered galaxy to recede (or to approach) but does result in the untethered galaxy joining the Hubble flow (v_pec → 0)." You say that the expansion does cause the galaxy to approach, so clearly you disagree with Tamara.

It is critical to recognize that the untethered galaxy does not "join the Hubble flow" because it is somehow immediately "carried away" from the origin by expansion. Instead, the galaxy actually accelerates toward the origin (counter to the expansion of the hypersphere), blasts through the origin, and carries its (coordinate negative) accumulated momentum out the opposite side. Thereafter the galaxy "joins the Hubble flow" only in the unremarkable sense that all peculiar velocities decay at 1/a: it keeps passing dust particles in the Hubble flow which happen to be moving faster and faster relative to its own proper velocity (as distance from the origin increases), and eventually the proper velocity of the nearby dust particles ahead of it is so great that the galaxy can't hardly pass them at all. All of this occurs though normal kinematics, without any need for an expanding hypersurface to add to or subtract from the galaxy's proper velocity.

4.1, 4.2, 4.3: I agree that the cosmic dust distribution and recession motion must be homogeneous and isotropic, the gravitational potential is the same everywhere, and there are no gradients. But gravitational effects must be felt inside a perfectly uniform gravitation field. It is the gravity of every dust particle which causes every other dust particle to accelerate toward it, slowing the original expansion. In the same way, the gravity of each dust particle causes a galaxy to move toward it.

4.5, 4.6: Imagine that our Milky Way galaxy is stationary in empty space, except that there is one identical galaxy at a certain distance to the "East" and another identical galaxy at the same distance to the "West." Our galaxy feels the same gravitational pull from both East and West, so it won't move and nothing happens, right? Wrong. All 3 galaxies will gravitationally attract each other, and after time passes the distances between them will decrease in proportion to the original separation. An observer on our galaxy says, "See, we haven't moved, the other galaxies approached us." But an observer on galaxy East says, "We haven't moved, and the Milky Way and Galaxy West both approached us!" Of course the question of who moved is a matter of reference frame, but everyone agrees that movement did occur (or that the universe shrank). Now add an infinite number of uniformly separated galaxies "Further East" and "Further West" along the East-West axis. This is a one dimensional model of a collapsing infinite FLRW universe. All of the galaxies will approach each other along the axis at rates proportional to their separation. Every galaxy will collapse toward every other galaxy. No matter how many additional galaxies we line up behind galaxy East, from our perspective galaxy East will never remain stationary or be pulled away from us, we always will see it as approaching us. That would be true even if we were the Western-most galaxy at the end of the string. Contrary to your statement in 4.6, the fact that evenly spread galaxies feel the same gravitational pull from every direction does not mean the gravity"cancels out".

4.4: I have found nothing in the literature which supports your notion that the Shell Theorem works only for a finite dust cloud. There are many references, but not one with that limitation. If the dust cloud were a just a thin shell around us, and then we progressively thicken that shell in infinitesimal increments, at what critical point does it become so thick that the mass of the shell external to the "critical thickness" suddenly begins affecting the cavity in the shell? That concept makes no sense, there is no such critical thickness.

In 1945 Einstein and Straus developed the Swiss-cheese model, which assumes that in fact the Shell Theorem (or Birkhoff's Theorem in relativity) can apply in an infinite FLRW universe. The Swiss-cheese model most commonly excavates a spherical cavity in an FLRW universe, compressing the matter in the cavity into a point mass. Einstein showed that the metric in the interior of the cavity is exactly Schwarzschild while the exterior metric remains FLRW. His purpose in this exercise was to describe a theoretical basis for why the expansion of the universe does not affect local orbits within our solar system or within our galaxy (causing them to become unstable). When the interior metric is exactly Schwarzschild, by definition the metric is governed entirely by the point mass, and mass exterior to the cavity has no effect. Although I haven't found an accessible copy of the Einstein-Straus article, I have read that Einstein used the same analogy of progressive infinitesimal increases of the thickness of the shell to help rationalize the Swiss-cheese model.

By the way, Birkhoff's Theorem says that a spherical region of dust can be treated as a point mass for gravitational purposes, without affecting the operation of the Schwarzschild metric. So when I have referred to that, I am not suggesting that the homogeneous distribution of dust in the region actually changes. I am just saying how the dust must be treated for gravitational calculations.

4.7: You quote Davis saying "The motion of this test galaxy reveals the effect the expansion of the Universe has on local dynamics." I agree 100%. The effect that the expansion has in accelerating the galaxy inward toward the origin is zero, as I already quoted from Davis. The expansion of the universe can be said (at the risk of some confusion) to have an effect on causing the galaxy to subsequently "join the Hubble flow", but only in the roundabout sense I already described.

Jon

Hi Jon, yea, at least we agree on something.

You wrote: "I think you're just saying that the galaxy has a peculiar velocity inward. I agree that its velocity is peculiar relative to the untethered galaxy's local Hubble flow at its own location, but it is not peculiar relative to the origin's local Hubble flow."

Careful with your terminology; peculiar velocity is always defined in terms of what we measure relative to the Hubble flow at the position of interest, never relative to us (except for the moment when the test galaxy eventually flies past us at close range).

Your: " If the peculiar velocity relative to the origin is zero, then by definition the galaxy cannot cross comoving coordinates in order to reach the origin, which is at a different comoving coordinate ."

Wrong definition of peculiar velocity, as stated above. It's only the proper velocity of the test galaxy that is initially zero. That proper velocity will change as soon as the expansion rate at the starting position changes, without any forces acting on the test galaxy. My 3.4 stands, I'm afraid. I will elaborate more in response to other statement below.

Your: " You are not distinguishing between comoving and proper motion here, so you're coming up with a mishmash (albeit a clever mishmash). "

I was not referring to comoving velocity here, just the movement of the galaxy through local space, as measured by a set of local comoving observers being passed. Only when the recession velocity exactly balances the local velocity will there be zero proper velocity. Only a (0,0) universe can maintain this condition.

Your: " If the proper velocity of a particle that starts with peculiar motion (relative to the background Hubble flow) actually increased as the Hubble rate drops (or even if the proper velocity remained constant), then peculiar velocities would not decay at exactly 1/a in comoving coordinates, the math would calculate a different result. So we know that any such notion is conclusively wrong. "

But it is only a negative proper (radial) velocity of the galaxy that increases in magnitude (getting more negative) as the Hubble rate drops. When it is going in any positive radial direction, the proper velocity decreases and peculiar velocity decays.

Your: " 3.5 In the absence of gravitational acceleration, the proper distance from the origin would never change, so this point is moot. "

I think you are dead wrong!

Your: " Tamara goes on to say (p.47): "Thus all terms in [dot R] cancel and we conclude that the expansion, [dot R] > 0 does not cause acceleration, [double dot D] > 0. Thus, the expansion does not cause the untethered galaxy to recede (or to approach) but does result in the untethered galaxy joining the Hubble flow (v_pec → 0)." You say that the expansion does cause the galaxy to approach, so clearly you disagree with Tamara. "

You are quoting (slightly) out of context from section (3-1.1), which is about "Expansion makes galaxies join the Hubble flow". This subsection only shows that any expansion, whether constant, accelerating or decelerating, makes galaxies with peculiar motion eventually join the Hubble flow. It appears that Barnes, Francis, James & Lewis (Sep 2006) dispute this analysis for the general case - espaceially the last paragraph on page 9. However, it is not a pear-reviewed paper and we will have to see how it stands up.

The strictly correct context is the in next section (3-1.2). Davis writes: " Thus the [uniform] expansion does not 'drag' the untethered galaxy away from us, even though the untethered galaxy does end up joining the Hubble flow. Only the acceleration of the expansion can result in a change in distance between us and the untethered galaxy. We have shown that the direction of that change is not always outwards. "

Clearly, change in expansion rate does the trick, as I have said before and still hold.

Have you contemplated how 'funny' a gravitational force (or gravitational acceleration) profile would be required in order to create the (1,0) curve of Davis' Fig. 3.2? The test galaxy 'almost immediately' (in cosmo-terms) acquires a proper velocity of ~ -0.05c and then essentially maintains that proper velocity for the next few hundred Gy, moving through the origin with no further appreciable acceleration or deceleration. If you can show me the math (or a reference) that produces that velocity profile from standard gravitational accelerations, I'll concede that your interpretation must be equally valid to mine. Until that time, I will consider it flawed.

As far as I can tell, the Peebles, Davis, etc. analyses and math are not based on gravitational accelerations according to the Shell Theorem. They are all based on the standard universal expansion considerations of the FLRW metric and Friedman equations.

-J

Hi Jorrie, I am eager to respond to your arguments, but you only responded to the first half of my note, so I need you to respond to the points I made in the second half. Otherwise those points will get lost in the shuffle.

In response to your statements about peculiar and proper velocity, I will offer this:

Let's say the galaxy is 1 mpc away from the origin when it is untethered, and the Hubble rate is 71 km/s/mpc. Thus the untethered galaxy's peculiar velocity is -71 km/s. At that rate, the untethered galaxy clearly is initially only treading water, in terms of proper distance relative to the origin, not getting any closer. From the galaxy's perspective, the origin is receding away from the galaxy at the same rate the galaxy is moving through the galaxy's local Hubble flow. In the absence of some "force" to impart proper acceleration, the proper distance will never decrease. The fact that comoving coordinates are expanding is irrelevant to the proper motion analysis, because the proper coordinates are not expanding.

Now you will argue that, from the galaxy's perspective, the origin's proper recession velocity away from it will decrease over time, as a result of the cosmic gravitational deceleration. (The obvious fallacy is that in your model gravity imparts acceleration to the dust but not to a galaxy moving through the dust). Let's do you one better and run a case where the origin's recession velocity drops to zero instantly upon the untethering. Then, at a proper velocity of 71 km/s toward the stationary origin, the unaccelerated galaxy would require 6.95E+13 years to reach the origin, that is, 69,500 GY. In that much time, the scale factor will have increased to ~ 200 a₀. And of course you are not even claiming that the recession velocity drops to zero immediately, so in your model the galaxy will require many times more than 69.5 trillion years to reach the origin. An unaccelerated proper approach velocity at that order of magnitude is just not going to generate the proper motion shown on the Davis or Barnes, Lewis charts.

Now let's switch to a peculiar velocity perspective. We know that the Hubble rate decays faster than the scale factor increases, and therefore faster than peculiar velocities decay (in comoving coordinates only) at 1/a. In fact, in an Ω=1 model, every time the scale factor doubles, the Hubble rate decreases by a factor of exactly 2^3/2 = 2.8284. Over the time interval when the galaxy's local peculiar velocity drops by half, the Hubble velocity drops by a factor of 2.8284, and so the local peculiar velocity of the unaccelerated galaxy (compared to the then-current Hubble velocity in its then-current neighborhood) does increase over time, and the galaxy's comoving velocity toward the origin increase. But at the same time, the comoving coordinates themselves are increasing their proper distance apart. This increase in the scale of the coordinates means that an increase in the peculiar velocity does not necessarily signify that the proper distance is decreasing.

One must recognize that the approach of the galaxy in comoving coordinates is to some degree just a mirage, because for example the scale factor of the coordinates increases at the same rate as the galaxy's comoving separation from the origin decreases. An example of that is shown by Davis' Fig 3.3 with the (0,0) trajectory, which asymptotically approaches the origin but never actually reaches it.

That's conceptually why its perfectly natual that the galaxy may not approach the origin at all in proper distance, in the absence of some means of proper acceleration.

Jon

Hi Jon, you wrote:

"Now you will argue that, from the galaxy's perspective, the origin's proper recession velocity away from it will decrease over time, as a result of the cosmic gravitational deceleration. (The obvious fallacy is that in your model gravity imparts acceleration to the dust but not to a galaxy moving through the dust)."

Where did you get the idea of the "fallacy"? In the hypersphere model, the momentums are present in the hyperspherical direction only for comoving particles and in both spatial and hyperspherical directions for particles with peculiar motion.

Your: " Let's do you one better and run a case where the origin's recession velocity drops to zero instantly upon the untethering. Then, at a proper velocity of 71 km/s toward the stationary origin, the unaccelerated galaxy would require 6.95E+13 years to reach the origin, that is, 69,500 GY. "

The "… origin's recession velocity drops to zero instantly upon the untethering" means that the whole hyperspherical motion da/dt is instantly changed to da/dt=0. This action imparts one huge, instant acceleration shock to all particles and will cause the untethered particle to acquire a huge negative proper velocity component. The broad idea is pictured (left) in the form of a portion of hypersphere, presented in one space and one hyperspace dimension. I just modified a picture from my eBook, so some information in black are superfluous.

χ is the comoving distance parameter and χR the proper distance of the tethered galaxy, where R is radius of curvature.

The observer and the distant object follow the red dotted hyperspace paths as scale factor a increases. While still tethered, the test galaxy follows the dotted blue path, at constant χR. The shock must obviously stop the observer and distant object in their hyperspace tracks and can be viewed as a hypervelocity change as indicated by the solid red vectors. The shock will however impart the same hypervelocity change to the test galaxy and will give it a huge negative proper velocity (solid blue vector), due to its peculiar hyperspace path when tethered.

There are many ways to describe this, but I prefer it in relation to the hypersphere model, because I can plot and visualize it for any expansion profile over time. If scale factor a increases as per the matter only (1,0) case, the deceleration of a_dot can be viewed as continuously imparting small shocks to the hypersphere and hence continuously accelerating the test galaxy towards the origin. This is just a more formal way of describing what I loosely stated in 3.1 to 3.6 of post #59.

This does mean that there is a form of proper acceleration working on the untethered galaxy, coming from large scale changes in expansion rate, not from a local Schwarzschild type of gravitational acceleration effect. I think this is at the core of our disagreement. I offer again: show me rigorously how such a local Schwarzschild gravity can produce the acceleration profiles, and I'll concede the point (at least halfway ).

-J

Hi Jorrie,

I'm sorry to put you to so much work charting out your "acceleration shock" idea but I think the problem is that I misstated my scenario in my post.

I talked about bringing the origin's proper recession velocity to zero, from the galaxy's perspective. My statement was nonsensical because the tether had already brought the origin's proper recession velocity to zero at the start of the exercise. As I had just described in the preceding paragraph!

What I should have said is that you would argue that, from the galaxy's perspective, the origin actually experiences proper acceleration toward the galaxy, as a result of the gravitational deceleration of the "expansion hypersphere". So the proper acceleration rate of the origin is equal to the deceleration rate of the universe.

Since in your model the proper distance between the galaxy and the origin is decreasing (at an increasing rate) over time, the total time required for the galaxy and the origin to move together would be a lot less than the 69.5 trillion years I calculated for a net 71 km/s approach rate. Maybe its even the same total time calculated by the kinetic model.

But I think you need to recognize that if you claim that the origin accelerates toward the galaxy in this scenario, then it is exactly equivalent to saying that the galaxy accelerates toward the origin. So in that sense you would have to admit that the gravitational acceleration of the galaxy which I have been describing in the kinetic model is a perfectly valid and accurate description of what is occuring.

If the results in the two models are the same, then one can claim that both are equally valid. It's really the same as the dichotomy between gravity attracting massive bodies vs. gravity curving spacetime which then causes massive bodies to change position.

So if this equivalence holds, I may have to admit that I couldn't actually "prove" that expanding space doesn't cause galaxies to separate. But you will need to admit that all of the kinematic principles I described (including the gravitational effect of an FLRW sphere) yield equally valid results.

Jon

Hi Jon, very interesting that we came to a similar realization on the possible equivalence of the two approaches without seeing each others post (submitted almost simultaneously) first! (see my reply 71 to S)

You wrote: "But I think you need to recognize that if you claim that the origin accelerates toward the galaxy in this scenario, then it is exactly equivalent to saying that the galaxy accelerates toward the origin."

From either the origin or the test galaxy perspective, the acceleration is towards each other, I agree. How else could it be? I'm still not sure that the kinematics of the interior Schwarzschild solution for a homogeneous dust sphere yields precisely the correct acceleration, as you claimed. Yes, it has the right profile, i.e. acceleration decreases as the test particle gets closer to the origin, but there is a time issue that bothers me.

In the interior Schwarzschild solution, clocks run slowest at the center, where the magnitude of the gravitational potential is lowest. I cannot see a way to reconcile this with the homogeneous, unbounded dust cloud where the gravitational potential is the same everywhere and hence static clocks all run at the same rate.

-J

Hi Jorrie,

I find it to be a very satisfactory outcome for the two approaches to be exactly equivalent. It certainly explains why professional experts have been able to debate the relative merits of the two perspectives for a long time with no clear resolution.

I have recently been shifting my view that gravitational time dilation should be calculated with the interior Schwarzschild solution. I now tend to think the exterior Schwarzschild solution must be the right answer. But I haven't been able to entirely dispense with the interior solution yet.

The exterior solution is definitely the one I prefer to apply, for one very powerful reason: In a spatially flat FLRW model, any arbitrarily defined homogeneous spherical region expands at exactly the Newtonian escape velocity of its total mass-energy contents. It can be easily shown that plugging the escape velocity formula into the SR time dilation formula causes an SR time dilation which is exactly equal to but opposite of the Schwarzschild exterior gravitational time dilation. So SR recession velocity and gravitational density can both contribute time dilation, but the net time dilation is zero!

This symmetry is too powerful to be a coincidence. In fact, this outcome is inevitable, because in the FLRW model, fundamental comoving observers all share exactly the same proper "cosmological time." Thus the FLRW model appears to inherently rule out any "net" time dilation of these types, in a light signal transmitted from one fundamental comoving observer to another.

By the way this is exactly the same symmetry that applies to an initially stationary particle infinitely distant from a black hole, which is gravitationally accelerated by the BH into a radially plunging trajectory. Because in this scenario the infalling particle's velocity is exactly equal to the BH's escape velocity, the particle's local region remains spatially flat, even after the particle plunges through the event horizon. And the particle's proper time remains exactly equal to the Schwarzschild coordinate time for distant observers. This can happen so close to a BH for one reason: in the Schwarzschild exterior metric, at escape velocity the gravitational time dilation exactly offsets the SR time dilation.

This symmetry alone doesn't provide a kinematic solution for the cosmological redshift, but it clears some major baggage out of the way so that the search for the real answer becomes more straightforward. I am pretty confident that the cosmological redshift will be shown to have equally valid solutions in both the "expanding hypersphere" and "kinematic" models; the equivalence principle motivates that outcome. And I think I know how to get there, but I haven't worked out the math yet...

Jon

Hi Jon,you wrote:

"I find it to be a very satisfactory outcome for the two approaches to be equivalent."

I'm not sure of the "exactly". You must remember that the approaches of Peacock, Tamara Davis, etc. are not the same as yours. They essentially use the Friedman equations, not a Schwarzschild solution.

"It can be easily shown that plugging the escape velocity formula into the SR time dilation formula causes an SR time dilation which is exactly equal to but opposite of the Schwarzschild exterior gravitational time dilation. So SR recession velocity and gravitational density can both contribute time dilation, but the net time dilation is zero!"

This is only approximately^[1] true, but I suppose in the weak field, slow moving regime, it is close enough. It is not true to say: "This can happen so close to a BH for one reason: in the Schwarzschild exterior metric, at escape velocity the gravitational time dilation exactly offsets the SR time dilation." Here you are talking of the strong field, fast moving regime.

Certainly an interesting approach and may work accurately enough for the low redshift region. I'm not sure it can work for the high redshift region. Or am I interpreting what you said wrongly?

-J

[1] I think you are ignoring the inevitable spatial curvature in the BH region that bedevils a SR treatment over any extended radial distances. Schwarzschild space can only be approximated as flat over very, very local regions. As I'm sure you know well, in geometrical units the proper time differential dτ of a particle plunging radially at exactly negative escape velocity Ve = -√[2M/r] at Schwarzschild radial parameter r from mass M is given by

dτ = √[1-2M/r - Ve²/(1-2M/r)] dt

where dt is the 'distant time' differential in asymptotically flat space. Calculate the 'rate of time' for two different radial distances and the results differ.

Hi Jorrie,

I did not make up the analysis that says space is locally flat for a particle plunging into a BH at escape velocity. I learned it from my favorite textbook, "Exploring Black Holes" by Edwin F. Taylor and John Archibald Wheeler (2000). They refer to the 'plunging at escape velocity reference frame' as the "rain frame" (analogous to free falling rain). They refer to stationary (orbiting) observers as if they were at rest on an orbiting mechanical "shell" built around the BH. They use questions (queries) to make many of their points.

In Query 3, p. B-7, they say:

"Shell separation as measured in the local rain frame. A local rain frame observer falls radially inward but has not yet reached the horizon. On the fly she measures the distance dr_rain between adjacent shells. According to special relativity, she measures the distance between adjacent shells to be shorter than the shell measurement, by the factor (1-v²)^1/2, where v is the relative speed between the in-faller and local shell. According to general relativity, the measured distance between adjacent shells is longer than the distance [measured by a distant observer] dr.... Show that these two factors cancel, so that (dr_rain/dr_shell)(dr_shell/dr) = dr_rain/dr = 1 and so we have the surprising result: dr_rain = dr. In other words, the in-falling observer measures the separation between adjacent shells to be equal to the difference dr in their Schwarzschild r-coordinate. From now on we can use dr instead of dr_rain. Note 1: [this] relation is true only for the rain observer, not observers falling slower or faster than the rain observer at a given radius. Note 2: [This relation] is correct only for separation between events that are simultaneous in the rain frame, as is true for measurement of length by any observer."

In Query 8, p. B-12, they say:

"Flat space in the rain frame? Set dt_rain = 0 and show that the space part of the [global rain frame metric] is what one would expect from Euclidean geometry. That is, the space part is locally flat. Is this result a surprise? In the Schwarzschild description, dr_shell is greater than dr outside the horizon. Why is there no similar stretch in the dr² term in the metric for the global rain frame? (Hint: Recall the results of Query 3.)"

They derive the following "global metric for the rain frame" (using c=1 simplification):

dτ = (1-2M/r)dt²_rain - 2(2M/r)^1/2 dt_raindr - dr² - r²dø²

They say that Charles Misner derived the form of this metric for the book, and that apparently this metric was originally derived by Paul Painleve and Allvar Gullstrand.

I misspoke when I said that the proper time coordinate of the plunging particle is the same as the time coordinate for a distant observer. For an infalling particle, the SR and gravitational time dilation are additive rather than canceling, so in effect there should be twice as much time dilation for the particle plunging at escape velocity as for a "shell" particle orbiting the BH.

Jon

Another oops in the last sentence of my last post. It should have read:

"... so in effect the time dilation is squared for the particle plunging at escape velocity as compared to a "shell" particle orbiting the BH."

I forgot that cumulative time dilation effects are multiplied together, not added together. The result is just what the global rain frame metric gives if you set dr = 0 and dø = 0.

Jon

Hi Jon.

I haven't read Edwin F. Taylor and John Archibald Wheeler (2000) (I'm still using Misner, Thorne and Wheeler). I do follow the "rain frame" and understand the local "flatness". However, I don't see how it could be used for a FLRW cosmic analysis, which is spatially flat until infinity, with common cosmic time.

As the untethered galaxy illustrates, the negative proper frame acceleration declines to zero as the galaxy passes us. Then, interestingly, it increases positively until it is about the same distance on the opposite side, where the acceleration starts to decline, tending to zero at infinity. It appears as if the prover velocity approaches a constant value on the opposite side. This acceleration profile is quite compatible with the interior solution, in fact a simple Newtonian interior (shell theorem) has precisely the right profile. I'm just not sure about the magnitude compatibility of the interior solution, but I definitively cannot see how it the FLRW can be compatible with the external solution.

What is compatible with the external solution though, is when all the mass-energy is concentrated at the center of the hypersphere and the cosmic dust sits on the expanding hypersurface. The hypersurface moves away from the center at escape velocity for the flat model. Further, all the dust is at equal potential and conforms to standard cosmic time. Something for you to think about.

-J

Hi Jorrie, you said:

"This acceleration profile is quite compatible with the interior solution, in fact a simple Newtonian interior (shell theorem) has precisely the right profile."

Please explain. Do you mean that the mathematical result of the interior solution is precisely right, or do you mean the heuristics are right?

The fact that all fundamentally comoving dust in the FLRW metric conforms to standard cosmic time is the cornerstone of my intuition that the Schwarzschild external metric and the gravitational time dilation exactly offset each other. How else could one explain the kinematic effects of gravity in a way that doesn't generate a net time dilation that violates the FLRW metric?

The hypersphere is a fine approach, but the equivalence principle says there must be a precisely equivalent analog in the kinematic model.

Jon

Hi Yon, on my statement and your question:

Me: "This acceleration profile is quite compatible with the interior solution, in fact a simple Newtonian interior (shell theorem) has precisely the right profile."

You: "Please explain. Do you mean that the mathematical result of the interior solution is precisely right, or do you mean the heuristics are right?"

A better analysis showed my statement to be wrong. I've plotted some more variables on the (now very busy) chart, i.e., proper velocity and proper acceleration of the test galaxy. Neither have the profile expected for either the exterior or the interior dynamics of Newton (or Schwarzschild for that matter). The magnitude of the acceleration drops rather rapidly to zero at D=0, goes only slightly positive and then drops away to zero.

This completely explains the "sudden kink" in the D-proper graph and also the almost linear movement after that, as is also apparent in Davis' figure 3.2.

An interior dynamics is expected to drop to zero more or less like the blue curve, but then to go positive with the same (mirror image) shape, unlike the curve. An exterior solution is expected the have increasing magnitude as distance decreases, in complete contradiction to the blue curve.

So what does this mean? Is it futile to try and do Schwarzschild kinematics in large scale cosmology? I don't know. You decide.

If you want to check the source of the data, download the updated spreadsheet. It has inevitably grown in size to about 1.3 MB in the process of getting data over longer timescales, creating more rows. I suppose VBA Macros will be a better solution, keeping the number of rows down without losing accuracy in the integration loop.

-J

Hi Jorrie,

Thanks, I downloaded your spreadsheet and I will play around with it.

I understand your concern about comparing Schwarzschild geodesics to FLRW geodesics. Of course, in Schwarzschild the point mass is constant and only the radius changes with time, while in FLRW both the radius and total mass encompassed by the radius change dynamically.

But that doesn't mean Schwarzschild is useless in modeling FLRW behavior, far from it. One must recognize that since total mass is constant in Schwarzschild, it can represent only a "snapshot" of a spherical FLRW region at a single instant in time. So one can plot a timelike geodesic only by taking a series of snapshots at regular intervals, and then plotting the changing acceleration vector and particle motion across snapshots, creating a "home movie". This accounts for the dynamically changing value of the total gravitating mass.

Naturally the geodesics in the home movie won't look the same as timelike Schwarzschild geodesics with a constant total mass. They're not supposed to -- they're supposed to look like FLRW geodesics.

A kinematic translation between instantaneous Schwarzschild accelerations and a dynamic FLRW acceleration must be mathematically perfect if one performs a mathematical integration of infinitesimal snapshot intervals. And the mere fact that the mass isn't actually stationary in the FLRW metric has no effect on its instantaneous gravitational potential. Birkhoff's Theorem confirms intuition on that point.

Jon

Hi again Jon.

Out of curiosity, I did a quick modeling of Davis' untethered galaxy, using the hyperspherical model in a spreadsheet. I get (roughly judged) the same results as Davis' fig. 3.2.

Below is a plot of the (1,0) CDM case, from D to the origin. The model also gives the "correct looking" results for the other cases, e.g. (0.3,0.7) LCDM. It seems to me to be orders of magnitude simpler than a kinematics model based upon Schwarzschild accelerations.

You can download the spreadsheet from my website if you want to have a look at it. It's not a "product", but strictly experimental...

-J

Hi Jorrie,

Hey that's a nice chart, thanks for running it.

Is it possible for you to do a version of the chart in comoving coordinates, and show how the (1,0) line compares to a line showing peculiar velocity decaying at 1/a? I'm wondering whether the accepted 1/a decay rate holds on an "inbound" trajectory toward the origin.

Also by the way, your updated cosmo calendar doesn't work very well with negative z values (i.e. the future). I'm trying to track the growth rate of a in the future and I can't do it that way.

It might be worthwhile adding a scale factor "a" calculation to the calculator.

Further by the way, conceptually does a(t) * "radius of observable universe_now" always equal the "radius of observable universe_then"? ("radius of observable universe_then" = "particle horizon_then") In other words, must the particle horizon at different historic times always encompass the same amount of ponderable matter as is encompassed within our present particle horizon? (Assuming no conversion of matter to radiation, etc.) The concepts of "a(t)" and "particle horizon_then" are generated from different starting points so perhaps they do not remain equal over time (???)

Jon

Hi Jon.

You asked: "Is it possible for you to do a version of the chart in comoving coordinates, and show how the (1,0) line compares to a line showing peculiar velocity decaying at 1/a? I'm wondering whether the accepted 1/a decay rate holds on an "inbound" trajectory toward the origin."

You made me curious as well, so I did put some extra calculations into the plot. I had to 'turn the graph', using time as the independent variable for obvious reasons. Surprisingly, V_peculiar follows the 1/a rule to the letter, even for the in-falling part and approaches zero (joining the Hubble flow) as time goes on. It doesn't look like it, but the Hubble flow also peters out to zero as time tends to infinity.

The lilac comoving distance curve is obviously for the test galaxy, visibly joining the Hubble flow (same as Davids' curve). The dark blue proper distance curve seems to just keep on going steadily. In reality, once it passes the -100 Mpc mark, it also curves very, very slowly and I think it must eventually approach the Hubble recession velocity.

I will discuss the cosmo-calculator separately.

-J

Hi Jorrie,

Outstanding graph! I was fairly confident that 1/a would prevail, but I wasn't sure. This is really helpful.

Jon

Hi Jorrie,

I keep coming back to the fact that your chart seems to confirm that peculiar velocity decays at 1/a on the "inbound" leg before it passes through the origin. I am sure that must be wrong. Think about it:

From a comoving perspective:

1. The untethered galaxy begins with an inward (negative) peculiar velocity of zero, while the local outward Hubble flow has positive velocity, meaning the galaxy's starting peculiar velocity is negative (as your graph shows).

2. Thereafter the galaxy moves through neighborhoods with increasingly smaller outward Hubble flows (because those neighborhoods are closer to the origin), so its inward negative peculiar velocity should be increasing steadily.

3. The outward Hubble rate decays slightly as time passes during the inbound leg, causing the untethered galaxy's inward velocity to pull even slightly more ahead of the local outward Hubble rate.

From a proper distance perspective:

1. The galaxy's proper velocity at first accelerates gravitationally rapidly in the inward (negative) direction, then remains pretty flat until the untethered galaxy passes through the origin.

2. Thereafter the galaxy moves inward through neighborhoods with increasingly smaller outward Hubble flows (because those neighborhoods are closer to the origin), so the local outward (positive) proper Hubble velocity decreases steadily relatively to the inward (negative) proper velocity of the galaxy, such that the disparity in proper velocity between them increases with time.

2. Gravitational acceleration toward the origin (in the negative direction) causes the local outward (positive) Hubble recession velocity at any fixed distance from the origin to decay with time. This works in the same direction as the gravitational acceleration of the galaxy toward the origin (in the negative direction), which causes the galaxy's negative velocity to increase slightly over time. This summation of accelerations causes the galaxy's inward proper velocity to increase even faster relative to the local outward proper Hubble flow.

I think your spreadsheet achieves the anomalous result it does because it calculates peculiar velocity based on an assumed 1/a decay rate, rather than as the absolute difference between then-current proper velocity and then-current proper Hubble flow, both as measured relative to the origin. So you are graphing a self-fulfilling prophesy.

Do you want to re-look at the spreadsheet?

Jon

Hi Jon, you wrote: "I think your spreadsheet achieves the anomalous result it does because it calculates peculiar velocity based on an assumed 1/a decay rate, rather than as the absolute difference between then-current proper velocity and then-current proper Hubble flow, ..."

No, I did not do that; I calculated peculiar velocity, but maybe from the wrong definition.

I used: V_pec = V_proper - H(t) D_proper.

It is probably more correct to use: V_pec = V_proper - H(t) D_comove, i.e., the simple subtraction of the Hubble flow from the proper velocity at any time.

However, this still gives just a different 1/a curve (green) for peculiar velocity, still starting as a large negative value, rapidly declining to equal the proper velocity at the origin. It becomes less negative than the proper velocity on the negative distance side (Hubble flow is negative), but eventually joins the proper velocity curve (I have checked that they converge).

The proper velocity reaches zero as time => infinity, which is the "asymptotic joining of the Hubble flow". It may not be a perfect 1/a curve, which is not required anyway, but it seems to me that a Friedman equation analysis does not agree with the way your kinematic insight leads you (if I understood correctly what you meant).

According tot my spreadheet, V_Hubble declines more rapidly with time than what V_proper goes negative, driving the negative V_pec towards V_proper in the (1,0) 'in-falling' phase. It also does so for the out-going phase, of course.

If I'm right, this was indirectly a very good input, eliminating an error anyway!

-J

Hi Jorrie,

OK, I see I got myself confused. A negative of a negative makes a positive, etc.

Thanks for taking a look. I think it was definitely worth double checking this. Your new version of the peculiar velocity line looks like I would expect it to look.

Apparently the effect of the change in the Hubble velocity with changing distance is always equal to or more significant than the effect of gravitational acceleration. Interesting.

Jon

Thanks for drawing my attention to it, Jon!

-J

Hi Jon, part 2.

"Also by the way, your updated cosmo calendar doesn't work very well with negative z values (i.e. the future)."

Yep, it was intended for the past. We might need a totally different calculator for the future. Also "a" in the present calculator adds little - it is just 1/(z+1).

"Further by the way, conceptually does a(t) * "radius of observable universe_now" always equal the "radius of observable universe_then"? ("radius of observable universe_then" = "particle horizon_then")"

No, "a(t) * "radius of observable universe_now" simply says how far the present particle horizon region was then, in proper distance. This is not the same as "particle horizon_then", which is how far light could have traveled up to "age then", taking into account the changing expansion rate.

The amount of matter inside the changing particle horizon depends on the expansion law. In a CDM (1,0) case, the amount of matter increases and in the LCDM (0.3,0.7) case, the amount of matter decreases over time, I think. So it's probably only in the (0,0) case that it could remain constant, but then it doesn't matter because there's no matter!

-J

Hi Jorrie,

Well too bad about the negative z's. If I need to rely on a calculator I'll just work in the past instead of the future.

Thanks for answering my question about a(t) and the particle horizon. Am I safe in assuming that "radius of the observable universe_then" in your calculator is a true calculation of the particle horizon_then, and not a derivation of a(t)?

One reason I'm interested in the particle horizon is that I believe it is the correct parameter to use in calculating gravitational time dilation at various historical epochs. And a(t) would not be useful for that purpose.

Jon

Your: "Am I safe in assuming that "radius of the observable universe_then" in your calculator is a true calculation of the particle horizon_then, and not a derivation of a(t)?"

Yes, this is how I understand "Hellfire's" original algorithm.

Your: "One reason I'm interested in the particle horizon is that I believe it is the correct parameter to use in calculating gravitational time dilation at various historical epochs."

I don't understand - why would there have been time dilation at historical epochs in a homogeneous universe?

-J

Hi Jorrie,

You asked "what gravitational time dilation are you trying to calculate?"

It's the same gravitational time dilation that I have said may be exactly offset and cancelled out by the SR time dilation resulting from recession at escape velocity. If the two dilations exactly offset, then they are perfectly consistent with FLRW's requirement of uniform proper cosmological time for fundamental comoving particles.

In playing around with gravitational time dilation calculations, it occurs to me that the only gravity that can affect a particle is the gravity within the future light cone of that particle starting at the BB (or end of inflation). The particle horizon_then is that light cone. It's an oddball light cone for the same reasons standard cosmology defines an oddball light cone for light travel: some of the gravity originates from particles with recession velocities above the speed of light (and above the speed of gravity), so the gravitational effect first recedes (in the observer's frame) and then eventually begins approaching the observer when the gravity crosses the Hubble Sphere. Just like in Davis Fig 1.1 with proper distance coordinates.

Jon

Hi Jon, you wrote: "... some of the gravity originates from particles with recession velocities above the speed of light (and above the speed of gravity), so the gravitational effect first recedes (in the observer's frame) and then eventually begins approaching the observer when the gravity crosses the Hubble Sphere. Just like in Davis Fig 1.1 with proper distance coordinates."

Careful! I do not believe that the 'teardrop' light cone in Davis Fig 1.1 applies to gravity per se. It applies to a photon emitted at t ~ 0 at the particle_horizon_then. The FLRW model has a 'static' (3D) gravitational field in existence after inflation that permeates all space at that time. Some areas cannot exchange information due to the particle horizon, but they are all at the same temperature and gravitational potential, barring the tiny (quantum?) perturbations. The 'speed of gravity' only applies to changes in the gravitational potential in 3D space, which happens 'everywhere at the same time' in FLRW cosmological time. No gravitons involved.

Anyway, I think we have now exhausted this deviation from the original topic of the thread and maybe we should rather get back to GRB photons.

It was an interesting detour, nevertheless!

-J

Hi Jon, I haven't replied to the second half of your post 60, because I think we must get the top part sorted by hook or by crook, without too much clutter from other issues. Nevertheless, I'll make some comments on the latter part.

Your: "But gravitational effects must be felt inside a perfectly uniform gravitation field. It is the gravity of every dust particle which causes every other dust particle to accelerate toward it, slowing the original expansion."

Obviously, but this does not mean that any of the galaxies moves towards some center, because there is no center in the cosmos. This is what I meant by "gravity cancels out" in my 4.6.

Your: "I have found nothing in the literature which supports your notion that the Shell Theorem works only for a finite dust cloud."

I think it is self-evident. Every particle accelerates towards every other particle in an infinite (or finite, closed) cosmos, so you cannot apply an acceleration vector to any one galaxy. It is quite different for the solid, bounded 3-d spherical dust cloud with a center of mass. There particles in radial directions accelerate away from each other, while particles lying in tangential direction from each other accelerate towards each other.

Your part on the Einstein and Straus Swiss-cheese model deals with the Schwarzschild metric of point masses inside cosmic cavities. I don't think this is applicable to the FLRW cosmic model.

But, let's first try to come to grips with the top part of your post and my reply to that...

-J

Hi Jorrie, a quick response:

Of course galaxies don't move gravitationally toward a "center of the universe", because there is no unique center. Every observer anywhere in an FLRW universe is equally entitled to treat itself as the center of the universe, and to observe the fact that the cosmic dust's gravitation cause everything in the universe to accelerate toward herself. There is absolutely nothing paradoxical about the fact that every observer in every location can make the same identical observation. This uniform acceleration toward the observer applies equally to the background dust and to particles with peculiar velocities.

If you are going to argue that our local geometry (for example, for our galaxy) is not essentially Schwarzschild, you are really going out on your own limb.

Again, the fact that Birkhoff's Thereom allows a spherical region of the dissolute FLRW dust cloud to be treated computationally as if it had been compressed to a point mass does not mean that the dust mass actually changes its physical characteristics of homogeneous dispersion and isotropic recession. I don't see the point in your continuing references to the fact that a "real" point mass would look different from an FLRW sphere. Of course it would. But a point mass' gravitational effect on the motion of a free particle at the radius of the sphere would be exactly the same as the gravitational effect of an FLRW sphere of equal radius and total mass.

Jon

Hi Jon, an even quicker response.

Your: "If you are going to argue that our local geometry (for example, for our galaxy) is not essentially Schwarzschild, you are really going out on your own limb."

Not what I said. Just that things like our galaxy and voids, which are essentially Schwarzschild, are not FLRW compatible. They represent inhomogeneities.

-J

PS: I'm retiring now - will be back in our early morning hours, 6-7 hours from now...

A correction. When I wrote: "There particles in radial directions accelerate away from each other, while particles lying in tangential direction from each other accelerate towards each other", I had in mind a spherical mass that has been collapsed to a point-mass at the center. In a non-collapsed cloud, all particles accelerate towards each other, with the acceleration the lowest for the more centrally located particles.

-J